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They discuss combinatorial explorations \+ using the " }{TEXT 289 5 "Maple" }{TEXT -1 65 " system for symbolic co mpution in conjunction with packages like " }{TEXT 290 10 "Combstruct " }{TEXT -1 2 ", " }{TEXT 291 4 "Gfun" }{TEXT -1 6 ", and " }{TEXT 292 5 "Mgfun" }{TEXT -1 303 ". The series appeared in three volumes, \+ Volume I (1996), Volume II (1997), and Volume III (2001), which are av ailable from the web site of the Algorithms Project. The present work sheet lists case studies sorted by application domains, and also point s to additional tutorials and package introductions." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Note that several of th ese worksheets have been produced with earlier releases of the " } {TEXT 293 7 "algolib" }{TEXT -1 96 " packages, and have not been updat ed. You may thus get different outputs when reproducing them." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Tutorials and Introductions" }}{EXCHG {PARA 15 "" 0 "" {HYPERLNK 17 "Introduction to the Combinatorial Structures Package." 2 "combstruct/grammars" "" }{TEXT -1 26 " A light introduction to " } {TEXT 257 10 "Combstruct" }{TEXT -1 170 ". There, you'll learn about \+ the basics of specifications, how to get counting sequences, and how t o use predefined structures (subsets, permutations, combinations, etc) ." }}{PARA 15 "" 0 "" {HYPERLNK 17 "Combinatorial Structures Package. " 2 "combstruct/sample_struct" "" }{TEXT -1 32 " Here's a simple coll ection of " }{TEXT 258 10 "Combstruct" }{TEXT -1 184 " examples showin g how to generate random trees, investigate the distribution of height by simulation, enumerate functional graphs, alcohols (!), necklaces, \+ expression trees, and so on." }}{PARA 15 "" 0 "" {HYPERLNK 17 "The Com bstruct Package, Generating Functions." 2 "combstruct/generating_funct ions" "" }{TEXT -1 31 " Starting with version 3.0 of " }{TEXT 259 10 "Combstruct" }{TEXT -1 120 ", it becomes possible to produce generatin g function equations and to solve some of them. Also, a new function, called " }{TEXT 260 10 "allstructs" }{TEXT -1 60 " for the exhaustive generation of structures has been added." }}{PARA 15 "" 0 "" {HYPERLNK 17 "Introduction to the Package Ore_algebra, A Package for L inear Operators and Skew Polynomials." 2 "Ore_algebra/intro" "" } {TEXT -1 236 " Several algorithms for integration and summation have \+ a natural description in terms of linear differential and difference o perators, which in turn are well described by skew (or Ore) polynomial s. This was the starting point for the " }{TEXT 262 11 "Ore_algebra" }{TEXT -1 9 " package." }}{PARA 15 "" 0 "" {HYPERLNK 17 "Introduction \+ to the Groebner Package." 2 "Groebner/intro" "" }{TEXT -1 6 " The " } {TEXT 261 8 "Groebner" }{TEXT -1 393 " package implements a general Bu chberger algorithm to deal with ideals of several types of multivariat e polynomials. Available features are: (i) calculations over complicat ed ground fields are possible with a single package; (ii) many term or ders; (iii) calculations of common invariants of ideals and varieties; (iv) facilities for change of orderings; (v) calculations with skew p olynomials." }}{PARA 15 "" 0 "" {HYPERLNK 17 "Introduction to the Mgfu n Package." 2 "Mgfun,intro" "" }{TEXT -1 6 " The " }{TEXT 294 5 "Mgfu n" }{TEXT -1 118 " package is intended for the symbolic manipulation o f a large class of special functions and combinatorial sequences (" } {XPPEDIT 18 0 "d;" "6#%\"dG" }{TEXT -1 131 "-finite and holonomic func tions), especially for their symbolic summation and integration. It i s a user-oriented interface to the " }{TEXT 295 8 "Holonomy" }{TEXT -1 39 " package, by avoiding the user to call " }{TEXT 296 8 "Holonomy " }{TEXT -1 5 " and " }{TEXT 297 8 "Groebner" }{TEXT -1 42 " directly. It is designed to be close to " }{TEXT 298 4 "gfun" }{TEXT -1 11 " i n spirit." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "General Functional ity" }}{EXCHG {PARA 15 "" 0 "" {HYPERLNK 17 "Analysis of Algorithms wi th Combstruct." 2 "combstruct/algorithm_analysis" "" }{TEXT -1 31 " S tarting with version 3.2 of " }{TEXT 263 10 "Combstruct" }{TEXT -1 87 ", it is possible to perform some simple complexity analyses of algori thms operating on " }{TEXT 264 10 "Combstruct" }{TEXT -1 29 " structur es in the spirit of " }{TEXT 265 3 "Luo" }{TEXT -1 1 "." }}{PARA 15 " " 0 "" {HYPERLNK 17 "Generating Marked Combstruct Grammars." 2 "combst ruct/mark1" "" }{TEXT -1 322 " Version 3.2 contains new functions pro ducing grammars with marked objects. This makes it possible to analys e means and variance of parameters of various combinatorial objects. \+ The basic mechanisms are described here together with examples like: c ycles in permutations, path length or leaves in binary trees, and so o n." }}{PARA 15 "" 0 "" {HYPERLNK 17 "More Examples of Marking Combstru ct Grammars." 2 "combstruct/mark2" "" }{TEXT -1 59 " This worksheet c ontinues to explore the possibilities of " }{TEXT 266 16 "combstruct[m ark]" }{TEXT -1 213 ". How far is the common ancestor of two nodes in a random binary tree? What is the average distance between two nodes ? This and a few other examples related to non-crossing configura tion s are to be found there." }}{PARA 15 "" 0 "" {HYPERLNK 17 "Attribute G rammars and Combinatorics." 2 "combstruct/attributes" "" }{TEXT -1 28 " This worksheet introduces " }{TEXT 299 10 "Combstruct" }{TEXT -1 410 " capability for describing properties of structures, like pathlen gth of trees, using attribute grammars. Since for some structures and algorithms it is possible to define a property corresponding to the c omplexity of the algorithm on the structure, these functions provide a nother mechanism of algorithm analysis. However, in this case there i s access to more statistical information, such as higher moments." }}} }{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Combinatorics" }}{EXCHG {PARA 15 "" 0 "" {HYPERLNK 17 "Enumeration of Planar Configurations in Combi natorial Geometry." 2 "autocomb/noncrossing_1" "" }{TEXT -1 64 " Ther e, starting with Euler's counting of triangulations of an " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 142 "-gon, many planar configurations can be counted, listed and r andomly generated, automatically. This sheet may serve as an entry point to the " }{TEXT 272 10 "Combstruct" } {TEXT -1 5 " and " }{TEXT 273 4 "Gfun" }{TEXT -1 10 " packages." }} {PARA 15 "" 0 "" {HYPERLNK 17 "Enumerating alcohols and other classes \+ of chemical moleculs, an example of Poly\341's theory." 2 "autocomb/al cohols" "" }{TEXT -1 85 " Classes of chemical compounds can be repres ented by combinatorial models using the " }{TEXT 274 10 "Combstruct" } {TEXT -1 9 " package." }}{PARA 15 "" 0 "" {HYPERLNK 17 "Balls and Urns , Etc." 2 "autocomb/balls_and_urns" "" }{TEXT -1 211 " Balls and urns models are basic in combinatorics, statistics, analysis of algorithms and statistical physics. We demonstrate here how their properties ca n be explored using most of the functionalities of the " }{TEXT 275 10 "Combstruct" }{TEXT -1 9 " package." }}{PARA 15 "" 0 "" {HYPERLNK 17 "Combinatorics of Non-Crossing Configurations." 2 "autocomb/noncros sing_2" "" }{TEXT -1 294 " Take points on a circle and con sider graph s based on these points such that no edges cross. A fairly complete t heory of these constrained random graphs can be developed. Planarity \+ entails a very strong combinatorial decomposability that is especially well suited to a detailed treatment by " }{TEXT 267 10 "Combstruct" } {TEXT -1 1 "." }}{PARA 15 "" 0 "" {HYPERLNK 17 "Constrained Permutatio ns and the Principle of Inclusion-Exclusion." 2 "autocomb/permutations " "" }{TEXT -1 33 " This worksheet is based on the " }{TEXT 268 10 "C ombstruct" }{TEXT -1 5 " and " }{TEXT 269 4 "Gfun" }{TEXT -1 318 " pac kages. It shows how to enumerate many classes of permutations with co nstraints on ``succession gaps'' (differences between consecutive elem ents). This covers many celebrated combinatorial problems (like ``ren contre'' or ``menage''). Generating functions, recurrences, and asymp totics are obtained automatically." }}{PARA 15 "" 0 "" {HYPERLNK 17 "R obustness in Random Interconnection Graphs." 2 "autocomb/random_graphs " "" }{TEXT -1 33 " This worksheet is based on the " }{TEXT 270 10 "C ombstruct" }{TEXT -1 5 " and " }{TEXT 271 4 "Gfun" }{TEXT -1 165 " pac kages. It shows how to characterize the trade-offs between the densit y of edges in a graph, its connectivity by short paths, and its robust ness to link failure." }}{PARA 15 "" 0 "" {HYPERLNK 17 "Monomer-Dimer \+ Tilings." 2 "autocomb/polymer_tilings" "" }{TEXT -1 253 " The number \+ of ways a square lattice can be tiled with unit squares and dominoes i s a combinatorial problem related to physical models of phase transiti on. This worksheet applies combstruct to finding bounds on the asympt otic behaviour of this number." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "Analytic Combinatorics" }}{EXCHG {PARA 15 "" 0 "" {HYPERLNK 17 "A \+ Problem in Statistical Classification Theory." 2 "autocomb/hierarchy_t rees" "" }{TEXT -1 136 " Classification theory makes use of classific ation trees. This worksheet explores properties of random classificat ion trees using the " }{TEXT 276 10 "Combstruct" }{TEXT -1 13 " packag e and " }{TEXT 277 5 "Maple" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {HYPERLNK 17 "Pollard's Rho Algorithm." 2 "autocomb/Pollard_algo" "" } {TEXT -1 101 " An efficient and simple technique used to find factors of integers. We show in this worksheet how " }{TEXT 278 10 "Combstru ct" }{TEXT -1 5 " and " }{TEXT 279 4 "Gfun" }{TEXT -1 162 " can be use d to analyze a realistic combinatorial model of the algorithm and thus derive a probabilistic complexity analysis of this algorithm and vari ants of it." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "Special Function s Manipulations" }}{EXCHG {PARA 15 "" 0 "" {HYPERLNK 17 "Gfun and the \+ AGM." 2 "autocomb/AGM" "" }{TEXT -1 144 " The arithmetic-geometric me an is related to hypergeometric functions. This relation and a genera lization of it are explored and proved using " }{TEXT 281 4 "gfun" } {TEXT -1 1 "." }}{PARA 15 "" 0 "" {HYPERLNK 17 "Variations on the Sequ ence of Ap\351ry Numbers." 2 "autocomb/Apery_numbers" "" }{TEXT -1 131 " Finding a second order recurrence satisfied by combinatorial nu mbers was a crucial step in Ap\351ry's proof of the irrationality of \+ " }{XPPEDIT 18 0 "Zeta(3)" "6#-%%ZetaG6#\"\"$" }{TEXT -1 259 ". In th is session, we exemplify an algorithmic method for symbolic summation \+ by rediscovering the recurrence on these numbers and proving a combina torial identity that they satisfy. We also derive an efficient calcul ation of the first hundreds of digits of " }{XPPEDIT 18 0 "Zeta(3)" "6 #-%%ZetaG6#\"\"$" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {HYPERLNK 17 "An \+ Integral of a Product of four Bessel Functions." 2 "autocomb/Bessel_in tegral" "" }{TEXT -1 31 " We illustrate the use of the " }{TEXT 280 5 "Mgfun" }{TEXT -1 155 " package on the computation of a closed form \+ for an integral from a recent research paper. This nice integral invo lves the four types of Bessel functions." }}{PARA 15 "" 0 "" {HYPERLNK 17 "Borel Resummation of Divergent Series Using Gfun." 2 "au tocomb/Borel_resummation" "" }{TEXT -1 452 " For some ``irregular sin gular'' problems coming from differential equations, there exist forma l power series solutions that are everywhere divergent. These power s eries turn out to make sense as asymptotic expansions of actual soluti ons. The Borel summation technique is used to recover convergent repr esentations for these actual functions solutions. For a fairly large \+ class of integrands, this technique leads to algorithmic calculations \+ using " }{TEXT 300 4 "gfun" }{TEXT -1 1 "." }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 11 "Asymptotics" }}{EXCHG {PARA 15 "" 0 "" {HYPERLNK 17 "Pa tterns in Words." 2 "autocomb/DNA" "" }{TEXT -1 50 " How likely is it that a specific pattern occurs " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 34 " times in a random word of length " }{XPPEDIT 18 0 "n" "6#%\"nG " }{TEXT -1 98 "? How does the probability of this event depend on th e specific pattern? This worksheet applies " }{TEXT 283 10 "Combstruc t" }{TEXT -1 5 " and " }{TEXT 284 4 "Gfun" }{TEXT -1 77 " to this prob lem which has a connection to questions from biology of the DNA." }} {PARA 15 "" 0 "" {HYPERLNK 17 "A Seating Arrangement Problem." 2 "auto comb/channel_allocation" "" }{TEXT -1 20 " What happens when " } {XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 334 " persons arrive at a lunche onette and are so unfriendly that no one wants to sit next to an alrea dy occupied seat? This worksheet explores properties of this problem \+ when people arrive randomly. (This also serves as a simplified model \+ of channel occupation in mobile communication.) The complete treatmen t is entirely based on the " }{TEXT 285 4 "Gfun" }{TEXT -1 13 " packag e and " }{TEXT 286 5 "Maple" }{TEXT -1 50 "'s capabilities in solving \+ differential equations." }}{PARA 15 "" 0 "" {HYPERLNK 17 "Staircase po lygons, a simplified model for self-avoiding walks." 2 "autocomb/self_ avoiding_walks" "" }{TEXT -1 193 " There, we count pairs of non-cross ing paths in integer lattices of dimensions 2, 3 and more. We also ge t numerical asymptotics by a connection method. This sheet makes inte nsive use of the " }{TEXT 287 4 "Gfun" }{TEXT -1 9 " package." }} {PARA 15 "" 0 "" {HYPERLNK 17 "Asymptotics of the Stirling Numbers of \+ the Second Kind." 2 "autocomb/Stirling_numbers" "" }{TEXT -1 443 " Th e asymptotics of the Bell numbers is a classical problem which is trad itionally treated by the saddle-point method. The asymptotic scale re quired to perform the computation is non-trivial, and variants of this problem such as the asymptotic behaviour of the average value or the \+ variance of the Stirling numbers involve an indefinite cancellation in this scale. This worksheet exemplifies the use of a recent algorithm on this problem. " }{TEXT 282 29 "[Based on experimental code.]" }}} }}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } "autocomb/self_ avoiding_walks" "" }{TEXT -1 193 " There, we count pairs of non-cross ing paths in integer lattices of dimensions 2, 3 and more. We also ge t nufallfals3QpdQiĦfami .mfamilyfamiliar!kfamou}far+K!kryfast?#'!ky}׵fatmenfattfavoritfbox+  Qpd֌J½Cfd'5<)HOS featur mxfebruarfedou#'feedxfermatfestoonfew7+#'!kxEfewer !kffmffff .mffffffmfgfhfi;4QyqĦSx E  autocomb autocomb autocomb} autocomb autocomb]algolibalgolibx autocombE autocomb autocombjalgolib autocombMAD autocombalgolibalgolib autocombalgolib autocomb autocomb+ autocomb#'algolib.MAD!>algolibAalgolib9DMAD] autocomb%fMADQialgolib!k autocomby autocomb7p{VERSION 5 0 "IBM INTEL LINUX" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Normal" -1 30 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 35 "" 0 1 104 64 92 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 " 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35 "Philippe Flajolet, January 10, 1998" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{2PARA 0 "" 0 "" {TEXT -1 98 "This worksheet explores v ariations of some classical problems in combinatorial analysis, like t he " }{TEXT 298 12 "derangement " }{TEXT -1 21 "problem (also called \+ " }{TEXT 303 9 "rencontre" }{TEXT -1 18 " problem), or the " }{TEXT 299 6 "menage" }{TEXT -1 81 " problem. The descriptions of these probl ems as borrowed from [Comtet, 1974] are:" }}{PARA 0 "" 0 "" {TEXT -1 8 " -- " }{TEXT 304 21 "Derangement/Rencontre" }{TEXT -1 297 ": If guests to a party leave their hats on hooks in the cloakroom, and gra b a hat at good luck when leaving, what is the probability that nobody gets back his own hat? The problem is equivalent to estimating the nu mber of permutations without fixed point, that is to say, without sing leton cycles." }}{PARA 0 "" 0 "" {TEXT -1 8 " -- " }{TEXT 305 6 "M enage" }{TEXT -1 54 ": What is the number of possible ways one can arr ange " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 116 " married couples ( =menages) around a table such that men and wo3men alternate but no woma n seats next to her husband?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 154 "These problems are in fact permutation enumera tion problems from classical combinatorial analysis. The constraints c onsidered concern, for a permutation " }{XPPEDIT 18 0 "sigma=sigma[1] ,`...`,sigma[n]" "6%/%&sigmaG&F$6#\"\"\"%$...G&F$6#%\"nG" }{TEXT -1 8 ", its \"" }{TEXT 312 15 "succession gaps" }{TEXT -1 54 "\", that is, differences between consecutive elements, " }{XPPEDIT 18 0 "sigma[i+1 ]-sigma[i]" ",&&%&sigmaG6#,&%\"iG\"\"\"\"\"\"F(F(&F$6#F'!\"\"" }{TEXT -1 64 ", The derangement problem corresponds to permutations such that " }{XPPEDIT 18 0 "sigma[i]-i<>0" "0,&&%&sigmaG6#%\"iG\"\"\"F'!\"\"\" \"!" }{TEXT -1 24 ", the menage problem to " }{XPPEDIT 18 0 "sigma[i]- i <> 0,1" "6$0,&&%&sigmaG6#%\"iG\"\"\"F(!\"\"\"\"!\"\"\"" }{TEXT -1 91 ". In the second case, the constraints on indices and values may al so be taken to be modulo " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 29 ", 4that is taken \"cyclically\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 64 "The symbolic method in enumerative combin atorics as well as the " }{HYPERLNK 17 "Combstruct" 2 "combstruct" "" }{TEXT -1 258 " package that implements it are based on the concept of decomposability. Combinatorial objects defined by \"constraints\" are thus not normally accessible to this framework. However, as shown in \+ this worksheet, the enumeration of various types of constrained " } {TEXT 261 12 "permutations" }{TEXT -1 37 " can be treated by a combina tion of " }{HYPERLNK 17 "Combstruct " 2 "combstruct" "" }{TEXT -1 2 " , " }{HYPERLNK 17 "Gfun" 2 "gfun" "" }{TEXT -1 69 ", and the Maple sys tem. One may either impose that all such gaps be " }{TEXT 300 6 "forc ed" }{TEXT -1 24 " to belong a finite set " }{XPPEDIT 18 0 "Omega" "I& OmegaG6\"" }{TEXT -1 50 ", or dually impose that all gaps have values \+ that " }{TEXT 301 7 "exclude" }{TEXT -1 14 " elements of " }{XPPEDIT 18 0 "Omega" "I&OmegaG6\"" }{5TEXT -1 41 ". Forcing gaps to belong to a finite set " }{XPPEDIT 18 0 "Omega" "I&OmegaG6\"" }{TEXT -1 64 " lead s to finite-state models, while the exclusion of gaps from " } {XPPEDIT 18 0 "Omega" "I&OmegaG6\"" }{TEXT -1 79 " builds on the finit e-state models models via an inclusion-exclusion argument." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "The general id ea of counting by inclusion-exclusion is to enumerate by generating fu nctions (GF's) objects with a number of " }{TEXT 313 24 "distinguished exceptions" }{TEXT -1 59 " to a set of constraints. The principle is as follows: If " }{XPPEDIT 18 0 "F(z,u)" "-%\"FG6$%\"zG%\"uG" }{TEXT -1 61 " is the bivariate generating function of such objects, where " }{XPPEDIT 18 0 "z" "I\"zG6\"" }{TEXT -1 18 " records size and " } {XPPEDIT 18 0 "u" "I\"uG6\"" }{TEXT -1 61 " records the number of dist inguished exceptions of some type " }{XPPEDIT 18 0 "Omega" "I&OmegaG6 \"" }{TEXT -1 60 ", then inclusion-exclusion [e.g., Comtet, 1974] p6 rovides " }{XPPEDIT 18 0 "F(z,-1)" "-%\"FG6$%\"zG,$\"\"\"!\"\"" } {TEXT -1 42 " as the univariate generating function of " }{XPPEDIT 18 0 "Omega" "I&OmegaG6\"" }{TEXT -1 21 "-free configurations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Here, we show c ases where so-called \"" }{TEXT 314 21 "permutation templates" }{TEXT -1 695 "\" can be described by finite-state mechanisms, so that a mult ivariate generating function of templates is directly accessible to co mbstruct. (A template describes a class of permutations by specifying \+ what happens at some places while others are free, which is rendered b y a \"dont-care\" symbol.) A specialization of the multivariate GF tha t involves a sign-change (for inclusion-exclusion) and an integral tr ansformation effected on an auxiliary variable (for \"filling\" the fr ee positions in templates and transforming them into permutations) yie lds a counting GF for the original problem. Generating functions, recu rrences, and numerical values7 can be obtained automatically in this fr amework." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 224 "This demonstration worksheet was originally inspired by works of \+ Kostas Hatzis (Patras) relative to edge-disjoint paths in random graph s and of Bruno Codenotti (Pisa) relative to computing permanents of ci rculants matrices. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 263 10 "References" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 " [Comtet, 1974] : L. Comtet, " }{TEXT 264 22 "Advanced Comb inatorics" }{TEXT -1 15 ", Reidel, 1974." }}{PARA 0 "" 0 "" {TEXT -1 38 " [EIS]: N. Sloane and S. Plouffe, " }{TEXT 265 37 "The Encyclo pedia of Integer Sequences" }{TEXT -1 23 ", Academic Press, 1995." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "This Mapl e worksheet is based on the current versions of the Maple packages " } {HYPERLNK 17 "Combstruct" 2 "combstruct" "" }{TEXT -1 5 " and " } {HYPERLNK 17 "Gfun" 2 "gfun" "" }{TEXT 270 1 " " }{TEXT -1 42 "8(for ve rsion V.4) that can be found under " }{TEXT 266 30 "http://www-rocq.in ria.fr/algo/" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "with(combstruct);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#71%+allstructsG%&countG%%drawG%)finishedG%'g feqnsG%)gfseriesG%(gfsolveG%,iterstructsG%%markG%'momentG%+nextstructG %,prog_gfeqnsG%.prog_gfseriesG%-prog_gfsolveG%)varianceG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "with(gfun);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7V%(LaplaceG%.algebraicsubsG%.algeqtodiffeqG%.algeqtose riesG%.algfuntoalgeqG%&borelG%.cauchyproductG%.diffeq*diffeqG%.diffeq+ diffeqG%2diffeqtohomdiffeqG%,diffeqtorecG%)guesseqnG%(guessgfG%0hadama rdproductG%0holexprtodiffeqG%)invborelG%,listtoalgeqG%-listtodiffeqG%0 listtohypergeomG%+listtolistG%.listtoratpolyG%*listtorecG%-listtoserie sG%5listtoseries/LaplaceG%1listtoseries/egfG%4listtoseries/lgdegfG%4li sttoseries/lgdogfG%1listtoseries/ogfG%4listtoseries/revegfG%4listtoser ies/revogfG%,maxdegcoeff9G%*maxdegeqnG%,maxordereqnG%,mindegcoeffG%*min degeqnG%,minordereqnG%*optionsgfG%,poltodiffeqG%)poltorecG%/ratpolytoc oeffG%(rec*recG%(rec+recG%,rectodiffeqG%,rectohomrecG%*rectoprocG%.ser iestoalgeqG%/seriestodiffeqG%2seriestohypergeomG%-seriestolistG%0serie storatpolyG%,seriestorecG%/seriestoseriesG" }}}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 87 "1. Permutations with \+ forbidden \"position gaps\" and the principle of Inclusion-Exclusion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "This s ection is devoted to the combinatorial basis on which this whole works heet relies. We define the notion of " }{TEXT 320 9 "templates" } {TEXT 323 1 " " }{TEXT -1 236 "that are simple combinatorial objects o ut of which constrained permutations can be built. An elementary integ ral transformation implements the inclusion-exclusion principle and le ads to generating functions of constrained permutations. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "The: set " } {XPPEDIT 18 0 "Omega" "I&OmegaG6\"" }{TEXT -1 16 " is a subset of " } {XPPEDIT 18 0 "\{0..d\}" "<#;\"\"!%\"dG" }{TEXT -1 6 " with " } {XPPEDIT 18 0 "d" "I\"dG6\"" }{TEXT -1 59 " being its maximal element. We propose to count the number " }{XPPEDIT 18 0 "P[n]" "&%\"PG6#%\"nG " }{TEXT -1 17 " of permutations " }{XPPEDIT 18 0 "sigma=sigma[1],sigm a[2]..sigma[n]" "6$/%&sigmaG&F$6#\"\"\";&F$6#\"\"#&F$6#%\"nG" }{TEXT -1 11 " such that " }{TEXT 258 4 "none" }{TEXT -1 22 " of the position gaps " }{XPPEDIT 18 0 "sigma[k]-k" ",&&%&sigmaG6#%\"kG\"\"\"F&!\"\"" }{TEXT -1 13 " belongs to " }{XPPEDIT 18 0 "Omega" "I&OmegaG6\"" } {TEXT -1 108 ". By inclusion-exclusion, we first enumerate permutation s with a certain number of distinguished exceptions." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "For this purpose, one n eeds to determine " }}{PARA 0 "" 0 "" {TEXT 315 5 " " }{XPPEDIT 18 0 "F[n,j]" "&%\"FG6$%\"nG%\"jG" }{TEXT -1 31 " = the number of per mutations " }{XPPEDIT 18 0 "sig;ma=sigma[1],sigma[2]..sigma[n]" "6$/%&s igmaG&F$6#\"\"\";&F$6#\"\"#&F$6#%\"nG" }{TEXT -1 11 " such that " } {XPPEDIT 18 0 "n-j" ",&%\"nG\"\"\"%\"jG!\"\"" }{TEXT -1 23 " of the p osition gaps " }{XPPEDIT 18 0 "sigma[k]-k" ",&&%&sigmaG6#%\"kG\"\"\"F& !\"\"" }{TEXT -1 43 " are distinguished and forced to belong to " } {XPPEDIT 18 0 "Omega" "I&OmegaG6\"" }{TEXT -1 22 ", while the remainin g " }{XPPEDIT 18 0 "j" "I\"jG6\"" }{TEXT -1 100 " positions may be oc cupied by arbitrary elements, as long as the permutation property is s atisfied." }}{PARA 273 "" 0 "" {INLPLOT "6F-%'CURVESG6#7'7$$\"\"\"\"\" !$!\"\"F*7$$\"\"#F*F+7$F.F*7$F(F*F'-F$6#7'F-7$$\"\"$F*F+7$F6F*F0F--F$6 #7'F57$$\"\"%F*F+7$F=F*F8F5-F$6#7$F5F?-F$6#7$F{TEXT -1 19 " = the coefficient " }{XPPEDIT 18 0 "[z^n*u^j" "7#*&)%\"zG%\"nG\"\"\")%\"uG% \"jGF'" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "G" "I\"GG6\"" }{TEXT -1 64 " represents the number of templates (domino placements) of size " } {XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 11 " that have " }{XPPEDIT 18 0 "j " "I\"jG6\"" }{TEXT -1 27 " don't-care symbols (hence " } {XPPEDIT 18 0 "n-j" ",&%\"nG\"\"\"%\"jG!\"\"" }{TEXT -1 54 " distingui shed occurrences of a position gap lying in " }{XPPEDIT 18 0 "Omega" " I&OmegaG6\"" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 62 " Then, by filling the don't-care symbols \+ in templates, one has" }}{PARA 277 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F[n,j]=G[n,j]*j!" "/&%\"FG6$%\"nG%\"jG*&&%\"GG6$F&F'\"\"\"-%*fac torialG6#F'F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 88 "Finally , by inclusion-exclusion, the number of permutations without any posit ion gap in " }{XPPEDIT 18 0 "Omega" "I&OmegaG6\"" }{TEXT -1 3 " is" }} {PARA 261 "" 0 "" {TEX?T -1 1 " " }{XPPEDIT 18 0 "P[n]=sum((-1)^(n-j)*G [n,j]*j!,j=0..n)" "/&%\"PG6#%\"nG-%$sumG6$*(),$\"\"\"!\"\",&F&\"\"\"% \"jGF.F0&%\"GG6$F&F1F0-%*factorialG6#F1F0/F1;\"\"!F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 50 "This is also expressible as an integral transform " }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sum(p[ n]*z^n,n=0..infinity)=int(G(-z,-t)*exp(-t),t=0..infinity)" "/-%$sumG6$ *&&%\"pG6#%\"nG\"\"\")%\"zGF*F+/F*;\"\"!%)infinityG-%$intG6$*&-%\"GG6$ ,$F-!\"\",$%\"tGF:F+-%$expG6#,$F " 0 "" {MPLTEXT 1 0 98 "domino:=proc(x,y,l) local j; seq([[x+j,y],[@x+j+1,y],[ x+j+1,y+1],[x+j,y+1],[x+j,y]],j=0..l-1); end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'dominoG:6%%\"xG%\"yG%\"lG6#%\"jG6\"F,-%$seqG6$7'7$,& 9$\"\"\"8$F49%7$,(F3F4F5F4F4F4F67$F8,&F6F4F4F47$F2F:F1/F5;\"\"!,&9&F4! \"\"F4F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "domino2:=proc (x,y,l,s) domino(x,y,l),[[x+s,y],[x+s+1,y+1]],[[x+s+1,y],[x+s,y+1]]; e nd;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(domino2G:6&%\"xG%\"yG%\"lG% \"sG6\"F+F+6%-%'dominoG6%9$9%9&7$7$,&F0\"\"\"9'F6F17$,(F0F6F7F6F6F6,&F 1F6F6F67$7$F9F17$F5F:F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 218 "plot([domino2(1,-1,3,2),domino2(2,-2,3,0),domino(3,-3,3),domino2( 4,-4,3,0),domino2(5,-5,3,2),domino(6,-6,3),domino2(7,-7,3,1)],scaling= constrained,axes=none,color=blue,labels=[`A permutation template`,``], thickness=3):" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 35 "2. Templates \+ and permutations with " }{XPPEDIT 18 0 "Omega" "I&OmegaG6\"" }{TEXT -1 23 "-exceptions (programme)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 3A56 "Templates are decomposable objects and t hus they can be specified in the Combstruct language. As it turns out, they are described by a finite state model that translates all the al lowed transitions between adjacent positions. This section provides t he main routines that compute the specification of templates associate d with a given set of position gaps." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "We consider templates with some positio ns marked that are elements of " }{XPPEDIT 18 0 "Omega" "I&OmegaG6\" " }{TEXT -1 41 " (exceptions) and \"don't-care\" positions." }}{PARA 258 "" 0 "" {TEXT -1 70 "Templates can be described by a finite state \+ device. For instance, if " }{XPPEDIT 18 0 "Omega=\{0,1\}" "/%&OmegaG<$ \"\"!\"\"\"" }{TEXT -1 44 ", we have a linear version of the classical " }{TEXT 262 14 "menage problem" }{TEXT -1 16 ": If the choice " } {XPPEDIT 18 0 "omega=1" "/%&omegaG\"\"\"" }{TEXT -1 4 " of " } {XPPEDIT 18 0 "omega " "I&omegaG6\"" }{TEXT -1 4 " in " }{XPPEDBIT 18 0 "Omega" "I&OmegaG6\"" }{TEXT -1 24 " has been made at stage " } {XPPEDIT 18 0 "k-1" ",&%\"kG\"\"\"\"\"\"!\"\"" }{TEXT -1 10 ", that is " }{XPPEDIT 18 0 "sigma[k-1]=k" "/&%&sigmaG6#,&%\"kG\"\"\"\"\"\"!\"\" F'" }{TEXT -1 75 " has been chosen, then the permutation property impl ies that one must have " }{XPPEDIT 18 0 "sigma[k] <> k" "0&%&sigmaG6#% \"kGF&" }{TEXT -1 29 ", that is to say, the choice " }{XPPEDIT 18 0 "o mega=0" "/%&omegaG\"\"!" }{TEXT -1 24 " is forbidden (at stage " } {XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 44 ") in a template immediately after (at stage " }{XPPEDIT 18 0 "k-1" ",&%\"kG\"\"\"\"\"\"!\"\"" } {TEXT -1 11 ") a choice " }{XPPEDIT 18 0 "omega=1" "/%&omegaG\"\"\"" } {TEXT -1 75 ". In other words, the language of templates is such that \+ a domino of type \"" }{XPPEDIT 18 0 "`-x`" "I#-xG6\"" }{TEXT -1 42 "\" cannot be followed by a domino of type \"" }{XPPEDIT 18 0 "`x-`" "I#x -G6\"" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "More generCally, a " }{TEXT 302 5 "state" }{TEXT -1 15 " is any subset " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "\{0..d-1\}" "<#;\"\"!,&%\"dG\"\"\"\"\"\"!\"\"" }{TEXT -1 37 " whose meaning is that the values in " }{XPPEDIT 18 0 "s" "I\"s G6\"" }{TEXT -1 104 " are unavailable as current position gaps, due to previous \"commitments\". Only don't cares or values of " }{XPPEDIT 18 0 "omega" "I&omegaG6\"" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "Omega" " I&OmegaG6\"" }{TEXT -1 26 " that are compatible with " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 169 " can be then be taken, and there is a t ransition to a new state that records the necessary information about \+ occupied positions. The finite state system thus comprises " } {XPPEDIT 18 0 "2^(d-1)" ")\"\"#,&%\"dG\"\"\"\"\"\"!\"\"" }{TEXT -1 76 " states, and the cardinality of the alphabet is equal to the cardinal ity of " }{XPPEDIT 18 0 "Omega" "I&OmegaG6\"" }{TEXT -1 38 " plus one \+ (for the don't care symbol)." }}{PARA 0 "" 0 "" {TEXT -1 0 "D" }}{PARA 0 "" 0 "" {TEXT -1 56 "This can be expressed by combstruct specificati ons. Let " }{XPPEDIT 18 0 "a[i]" "&%\"aG6#%\"iG" }{TEXT -1 53 " be a l etter that represents a position gap equal to " }{XPPEDIT 18 0 "i" "I \"iG6\"" }{TEXT -1 5 ". An " }{XPPEDIT 18 0 "Omega" "I&OmegaG6\"" } {TEXT -1 20 " template of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" } {TEXT -1 39 " is thus described by a word of length " }{XPPEDIT 18 0 " n" "I\"nG6\"" }{TEXT -1 19 " over the alphabet " }{XPPEDIT 18 0 "A" "I \"AG6\"" }{TEXT -1 18 " formed with the " }{XPPEDIT 18 0 "a[omega]" " &%\"aG6#%&omegaG" }{TEXT -1 9 " for all " }{XPPEDIT 18 0 "omega" "I&om egaG6\"" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "Omega" "I&OmegaG6\"" } {TEXT -1 25 " and a don't care symbol." }}{PARA 0 "" 0 "" {TEXT -1 26 "First build the alphabet:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "build_alphabet:=proc(Omega) local omega; seq(a[omega]=Atom,omega= Omega),dontcare=Prod(Atom,marked), marked=Epsilon end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/buildE_alphabetG:6#%&OmegaG6#%&omegaG6\"F*6%-%$s eqG6$/&%\"aG6#8$%%AtomG/F39$/%)dontcareG-%%ProdG6$F4%'markedG/F<%(Epsi lonGF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "build_alphabet( \{0,1,3\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'/&%\"aG6#\"\"!%%AtomG/& F%6#\"\"\"F(/&F%6#\"\"$F(/%)dontcareG-%%ProdG6$F(%'markedG/F6%(Epsilon G" }}}{PARA 0 "" 0 "" {TEXT -1 54 "Next, build the transitions. The a uxiliary procedure " }{XPPEDIT 18 0 "MinusOne" "I)MinusOneG6\"" } {TEXT -1 40 " decreases all the elements of a set by " }{XPPEDIT 18 0 "1" "\"\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "MinusOne:=proc(S) map(proc(x) x-1 end,S) minus \{-1\} end; " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%)MinusOneG:6#%\"SG6\"F(F(-%&minusG6$ -%$mapG6$:6#%\"xGF(F(F(,&9$\"\"\"!\"\"F4F(F(F3<#F5F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 113 "The transitions (in a raw form) are thus given by t he compatibility relations between each state and the symbols." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "build_transition:=proc(sFtat e,Omega) local x,t; if state=\{\} then t:=Epsilon else t:=NULL fi;s[st ate]=Union(t,Prod(dontcare,s[MinusOne(state)]),seq(Prod(a[x],s[MinusOn e(state union \{x\})]),x=Omega minus state)) end;\n" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%1build_transitionG:6$%&stateG%&OmegaG6$%\"xG%\"tG6 \"F,C$@%/9$<\">8%%(EpsilonG>F3%%NULLG/&%\"sG6#F0-%&UnionG6%F3-%%ProdG6 $%)dontcareG&F96#-%)MinusOneGF:-%$seqG6$-F?6$&%\"aG6#8$&F96#-FE6#-%&un ionG6$F0<#FN/FN-%&minusG6$9%F0F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "build_transition(\{\},\{0,1\},1);build_transition(\{0 \},\{0,1\},1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"sG6#<\"-%&Union G6&%(EpsilonG-%%ProdG6$%)dontcareGF$-F-6$&%\"aG6#\"\"!F$-F-6$&F36#\"\" \"&F%6#<#F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"sG6#<#\"\"!-%&Unio nG6$-%%ProdG6$%)dontcareG&F%6#<\"-F-6$&%\"aG6#\"\"\"F$" }}}{PARA 0 "" 0 "" {TEXT -1 138 "The grammar is then obtained by collecting transiti ons and taking into account the don't care symbol. The initial state \+ is the empty set " }{XPPEDIGT 18 0 "\{\}" "<\"" }{TEXT -1 65 " and it i s also the final state (since no domino can \"protrude\")." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "build_grammar:=proc(Omega) [s[\{\} ],\{build_alphabet(Omega)\} union map(build_transition,combstruct[alls tructs](Subset(\{$0..max(op(Omega))-1\})),Omega),unlabelled] end;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%.build_grammarG:6#%&OmegaG6\"F(F(7%& %\"sG6#<\"-%&unionG6$<#-%/build_alphabetG6#9$-%$mapG6%%1build_transiti onG-&%+combstructG6#%+allstructsG6#-%'SubsetG6#<#-%\"$G6#;\"\"!,&-%$ma xG6#-%#opGF4\"\"\"!\"\"FOF5%+unlabelledGF(F(" }}}{PARA 0 "" 0 "" {TEXT -1 19 "Here is an example:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "build_grammar(\{0,1\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%& %\"sG6#<\"<(/F$-%&UnionG6&%(EpsilonG-%%ProdG6$%)dontcareGF$-F/6$&%\"aG 6#\"\"!F$-F/6$&F56#\"\"\"&F%6#<#F7/F=-F+6$F.F8/%'markedGF-/F4%%AtomG/F :FF/F1-F/6$FFFD%+unlabelledG" }}}{PARA 0 "" 0 "" {TEXT -1 34 "In gener al, the grammar comprises " }{XPPEDIT 18 0 "2^(d-1)" ")\"\"#H,&%\"dG\" \"\"\"\"\"!\"\"" }{TEXT -1 134 " nonterminal, which corresponds to a f inite-stae device with as many states. The template GF's are thus rat ional functions of degree " }{XPPEDIT 18 0 "2^(d-1)" ")\"\"#,&%\"dG\" \"\"\"\"\"!\"\"" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "d" "I\"dG6\"" }{TEXT -1 27 " is the maximal element of " }{XPPEDIT 18 0 "Omega" "I&O megaG6\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 63 "3. Special cases of permutations with for bidden \"position gaps\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 97 "In this section, we enumerate various classes of p ermutations associated with a given finite set " }{XPPEDIT 18 0 "Omega " "I&OmegaG6\"" }{TEXT -1 193 " of forbidden position gaps. The proces s is in three steps: (i) Generate the grammar by means of the procedur es of Section 2; (ii) compute automatically a bivariate GF of template s by means of " }{HYPERLNK 17 "Combstruct[gfsolve]" 2 "combstruct[gfso lve]" "" }{TEXT -1 I493 "; (iii) transform the corresponding GF by mean s of the integral transform of Section 1 that implements the inclusion -exclusion principle.All cases can be treated automatically, and we de tail here the derangement and menage problems. Several generalization s of the menage problem are also tabulated. The ordinary GF's turn out to have hypergeometric forms and satisfy simple (but combinatorially \+ nonobvious) linear recurrences of the holonomic type that can once mor e be derived automatically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 28 "Derange ments (or Rencontres)" }}{PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 267 11 "derangement" }{TEXT -1 79 " [Comtet, 1974] is a permutation withou t fixed point, that is to say, the set " }{XPPEDIT 18 0 "Omega=\{0\} " "/%&OmegaG<#\"\"!" }{TEXT -1 94 ". of position gaps is forbidden. A \+ derangement is thus a permutation without singleton cycles." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "G0:=build_Jgrammar(\{0\},0);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%#G0G7%&%\"sG6#<\"<&/F&-%&UnionG6%%(E psilonG-%%ProdG6$%)dontcareGF&-F16$&%\"aG6#\"\"!F&/%'markedGF//F6%%Ato mG/F3-F16$F=F;%+unlabelledG" }}}{PARA 0 "" 0 "" {TEXT -1 79 " The lang uage of templates is thus just the set of all words over the alphabet \+ " }{XPPEDIT 18 0 "\{a[0],`*`\}" "<$&%\"aG6#\"\"!%\"*G" }{TEXT -1 7 " w here " }{XPPEDIT 18 0 "`*`" "I\"*G6\"" }{TEXT -1 38 " is shorthand for a don't-care symbol." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dra w(G0,size=10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%%ProdG6$&%\"aG6#\" \"!-F$6$-F$6$%%AtomG%'markedG-F$6$F,-F$6$F&-F$6$F,-F$6$F&-F$6$F&-F$6$F ,-F$6$F,-F$6$F&%(EpsilonG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "seq(count(G0,size=n),n=0..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6- \"\"\"\"\"#\"\"%\"\")\"#;\"#K\"#k\"$G\"\"$c#\"$7&\"%C5" }}}{PARA 0 "" 0 "" {TEXT -1 17 "The bivariate GF " }{XPPEDIT 18 0 "G(z,u)" "-%\"GG6$ %\"zG%\"uG" }{TEXT -1 36 " is obtained by combstruct[gfsolve]:" }} {EXCHG {KPARA 0 "> " 0 "" {MPLTEXT 1 0 42 "gfsolve(op(2,G0),op(3,G0),z, [[u,marked]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/-%)dontcareG6$%\" zG%\"uG*&F(\"\"\"F)F+/-&%\"sG6#<\"F',$*$,(!\"\"F+F*F+F(F+F5F5/-&%\"aG6 #\"\"!F'F(/-%'markedGF'F)" }}}{PARA 0 "" 0 "" {TEXT -1 15 " In particu lar," }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "g0:=subs(\",s[\{\}]( z,u));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g0G,$*$,(!\"\"\"\"\"*&%\" zGF)%\"uGF)F)F+F)F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 107 "The counting s equence is obtained by the transformation corresponding to the inclusi on-exclusion principle." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "s eries(subs([z=-z,u=-t],g0),z=0,11):map(proc(x) int(x*exp(-t),t=0..infi nity) end,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+9%\"zG\"\"\"\"\"!F% \"\"#\"\"#\"\"$\"\"*\"\"%\"#W\"\"&\"$l#\"\"'\"%a=\"\"(\"&L[\"\"\")\"'' \\L\"\"\"*\"(h\\L\"\"#5-%\"OG6#F%\"#6" }}}{PARA 0 "" 0 "" {TEXT -1 13 "The sequence " }{XPPEDIT 18 0 "1,0,1,2,9,44,265,`...`" "6*\"\"\"\"\"! \"\"\"\"\"#\"\"*\"#W\"$l#%$...G" }L{TEXT -1 32 ", is of course classica l. It is " }{TEXT 271 5 "M1937" }{TEXT -1 72 " of [EIS] where it is de scribed as \"subfactorial or rencontre numbers\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "The ordinary generating f unction (OGF) is also accessible in closed-form as" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 52 "P0:=int(exp(-t)*subs([z=-z,u=-t],g0),t=0..in finity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P0G,$*(-%#EiG6$\"\"\",$ *&,&%\"zGF*F*F*F*F.!\"\"F/F*F.F/-%$expG6#F+F*F/" }}}{PARA 0 "" 0 "" {TEXT -1 39 "This involves the exponential integral " }{XPPEDIT 18 0 " Ei" "I#EiG6\"" }{TEXT -1 45 ", where the exponential integral symbol m eans" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 275 "" 0 "" {XPPEDIT 18 0 "Ei(1,x)=int(exp(-xt)/t,t=1..infinity)" "/-%#EiG6$\"\"\"%\"xG-%$i ntG6$*&-%$expG6#,$%#xtG!\"\"\"\"\"%\"tGF1/F3;\"\"\"%)infinityG" }} {PARA 0 "" 0 "" {TEXT -1 96 "The OGF is purely divergent and P0 is to \+ be interpreted as asymptotic to the right-hand side as " }{XPPEDITM 18 0 "z" "I\"zG6\"" }{TEXT -1 10 " tends to " }{XPPEDIT 18 0 "0^`-`" ")\" \"!%\"-G" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " map(simplify,asympt(subs(z=-1/y,P0),y,10));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,4\"\"\"F$*$%\"yG!\"#F$*$F&!\"$F'*$F&!\"%\"\"**$F&!\"&! #W*$F&!\"'\"$l#*$F&!\"(!%a=*$F&!\")\"&L[\"-%\"OG6#*$F&!\"*F$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 28 "Verifica tion with Combstruct" }{TEXT -1 153 ". Derangements are characterized \+ by their cycle decomposition as labelled sets of cycles with length at least 1. They are thus specifiable in combstruct." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 43 "der:=[S,\{S=Set(Cycle(Z,card>1))\},labelled] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$derG7%%\"SG<#/F&-%$SetG6#-%&Cy cleG6$%\"ZG2\"\"\"%%cardG%)labelledG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(count(der,size=n),n=0..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-\"\"\"\"\"!F#\"\"#\"\"*\"#W\"$l#\"%a=\"&L[\"\"''\\L\"\" (h\\L\"" }}}{EXCHG {PARA 0 "> " 0 "N" {MPLTEXT 1 0 31 "gfsolve(op(2,der ),op(3,der),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/-%\"SG6#%\"zG,$* &-%$expGF'!\"\",&F-\"\"\"F(F/F-F-/-%\"ZGF'F(" }}}{PARA 0 "" 0 "" {TEXT -1 56 " The exponential generating function (EGF) is well-known " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "der_egf:=simplify(subs( \",S(z)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(der_egfG,$*&-%$expG6# ,$%\"zG!\"\"\"\"\",&F,F-F+F-F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "series(der_egf,z=0,13);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+=%\"zG\"\"\"\"\"!#F%\"\"#\"\"##F%\"\"$\"\"$#F+\"\")\"\"%#\"#6\" #I\"\"&#\"#`\"$W\"\"\"'#\"$.\"\"$!G\"\"(#\"%>@\"%gd\"\")#\"&(o;\"&g`% \"\"*#\"&\"[;\"&+[%\"#5#\"(d%o9\"(!o\"*R\"#6#\")J&>g\"\")+caV\"#7-%\"O G6#F%\"#8" }}}{PARA 0 "" 0 "" {TEXT -1 42 "We check that the coefficie nts in the EGF " }{XPPEDIT 18 0 "der_egf" "I(der_egfG6\"" }{TEXT -1 16 " and in the OGF " }{XPPEDIT 18 0 "P0" "I#P0G6\"" }{TEXT -1 67 " ar e the same. (The two GF's are related by the Laplace transform.)" }} {PARA 0 ""O 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 277 11 "Asymptot ics" }{TEXT -1 37 ". The number of derangements of size " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 45 " satisfies the well-known asymptotic formula:" }}{PARA 262 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p[n]/n!= exp(-1)*(1+o(1))" "/*&&%\"pG6#%\"nG\"\"\"-%*factorialG6#F'!\"\"*&-%$ex pG6#,$\"\"\"F,F(,&\"\"\"F(-%\"oG6#\"\"\"F(F(" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 62 "a fact which is obvious given the singularity o f the EGF at 1:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "series(de r_egf,z=1,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++,&!\"\"\"\"\"%\"zGF &,$-%$expG6#F%F%!\"\"F)\"\"!,$F)#F%\"\"#\"\"\"-%\"OG6#F&\"\"#" }}}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 14 "Menage numbers" }}{PARA 0 "" 0 " " {TEXT -1 4 "The " }{TEXT 273 14 "menage numbers" }{TEXT -1 34 " are \+ defined by the excluded set " }{XPPEDIT 18 0 "Omega=\{0,1\}" "/%&Omeg aG<$\"\"!\"\"\"" }{TEXT -1 354 ". In nineteenth century terminology, t his is phrased as follows: In how many wPays can one place couples with men and women alternating in such a way that no husband can seat next to his wife? The problem is usually posed in terms of a round table ( =cyclic permutations); see [Comtet, 1974]. In this section, the linea r version of the problem is treated." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "G01:=build_grammar(\{0,1\},1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$G01G7%&%\"sG6#<\"<(/%'markedG%(EpsilonG/&%\"aG6#\"\" !%%AtomG/%)dontcareG-%%ProdG6$F3F,/&F06#\"\"\"F3/&F'6#<#F2-%&UnionG6$- F76$F5F&-F76$F:F>/F&-FB6&F-FD-F76$F/F&FF%+unlabelledG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "g01:=subs(gfsolve(op(2,G01),op(3,G0 1),z,[[u,marked]]),s[\{\}](z,u));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %$g01G,$*&,&!\"\"\"\"\"%\"zGF)F),*F)F)*&F*F)%\"uGF)F(F*!\"#*$F*\"\"#F) F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 78 " The linear \"menage numbers\" i mmediately result by the general transformation." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 92 "series(subs([z=-z,u=-t],g01),z=0,17):ser01:=ma p(proQc(x) int(x*exp(-t),t=0..infinity) end,\");" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&ser01G+C%\"zG\"\"\"\"\"!F'\"\"$\"\"$\"\"%\"#;\"\"&\" #'*\"\"'\"$v'\"\"(\"%8a\"\")\"&+)[\"\"*\"'#f)[\"#5\"(L$z`\"#6\")vffk\" #7\"*)G#>S)\"#8\",_nin<\"\"#9\"-ND1ul<\"#:\".$fJb'f#G\"#;-%\"OG6#F'\"# <" }}}{PARA 0 "" 0 "" {TEXT -1 17 "This is sequence " }{TEXT 324 5 "M3 020" }{TEXT -1 5 " of [" }{TEXT 325 3 "EIS" }{TEXT -1 45 "], defined t here as \"sums of menage numbers\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 326 8 "Checking" }{TEXT -1 125 ". Combstruct is also useful for checking and debugging purposes. Consistency of templ ates can be checked for small values of " }{XPPEDIT 18 0 "n" "I\"nG6\" " }{TEXT -1 22 " by expansions and by " }{TEXT 274 22 "combstruct[alls tructs]" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "s eries(g01,z=0,10):map(expand,\");" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+ 9%\"zG\"\"\"\"\"!,&%\"uGF%F%F%\"\"\",(*$F(\"\"#F%F(\"\"$F%F%\"\"#,**$F (F-F%F+\"\"&F(\"\"'F%F%\"\"$,,F0\R"\"(F+\"#:F(\"#5F%F%*$F(\"\"%F%\"\"%, .F8\"\"*F0\"#GF+\"#NF(F6*$F(F1F%F%F%\"\"&,0F0\"#%)F+\"#qF(\"#@F%F%F8\" #XF?\"#6*$F(F2F%\"\"',2F8\"$l\"F0\"$5#F+\"$E\"F(F=F?\"#mF%F%FG\"#8*$F( F5F%\"\"(,4F0\"$i%F+FKF(\"#OF%F%F8\"$&\\F?\"$'GFG\"#\"*FOF6*$F(\"\")F% \"\"),6F8\"%(G\"F0\"$C*F+\"$I$F(FEF?\"%,5F%F%FG\"$b%FO\"$?\"FW\"#<*$F( F " 0 "" {MPLTEXT 1 0 46 " allstructs(G01,size=2);allstructs(G01,size=3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7'-%%ProdG6$-F%6$%%AtomG%'markedG-F%6$F'%(EpsilonG-F%6$ F'-F%6$&%\"aG6#\"\"!F--F%6$&F36#\"\"\"F+-F%6$F2F0-F%6$F2F+" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7/-%%ProdG6$&%\"aG6#\"\"\"-F%6$F'-F%6$-F%6$% %AtomG%'markedG%(EpsilonG-F%6$F/-F%6$&F(6#\"\"!F--F%6$F/-F%6$F8-F%6$F8 F3-F%6$F'-F%6$F/F?-F%6$F8-F%6$F/F--F%6$F/F+-F%6$F/FG-F%6$F8F+-F%6$F8F6 -F%6$F/FC-F%6$F8FC-F%6$F'FG-F%6$F8F=" }}}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT 275 15 "Holonomic forms" }{TEXT -1 47 ". A r ecurrence can be heuristically guessed by " }{XPPEDIT 18 0 "gfSun" "I%g funG6\"" }{TEXT -1 139 ", proved by the hypergeometric method detailed here (this is however specific to the problem at hand), and also obt ained automatically by " }{XPPEDIT 18 0 "Mgfun" "I&MgfunG6\"" }{TEXT -1 57 ". Problems of this type a priori belong to the so-called " } {TEXT 268 15 "holonomic class" }{TEXT -1 207 ", introduced by Zeilberg er, that is to say, sequences that satisfy linear recurrences with pol ynomial coefficients. Alternatively, the GF's satisfy linear different ial equations with polynomial coefficients." }}{PARA 0 "" 0 "" {TEXT -1 35 "As a first approach, the procedure " }{HYPERLNK 17 "gfun[listto diffeq]" 2 "gfun[listtodiffeq]" "" }{TEXT -1 86 " provides a plausible differential equation that accounts for a given number sequence." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "l01:=[seq(coeff(ser01,z,i),i =0..16)];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$l01G73\"\"\"\"\"!F'F& \"\"$\"#;\"#'*\"$v'\"%8a\"&+)[\"'#f)[\"(L$z`\")vffk\"*)G#>S)\",_nin<\" \"-ND1ul<\".$fJb'f#G" }}T}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ode 01:=listtodiffeq(l01,Y(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode0 1G7$<$,*\"\"\"F(*$%\"zG\"\"#!\"\"*&,(F,F(F)F(*$F*\"\"$F(F(-%\"YG6#F*F( F(*&,&F/F(F)F(F(-%%diffG6$F1F*F(F(/-F26#\"\"!F(%$ogfG" }}}{PARA 0 "" 0 "" {TEXT -1 77 "This suggests a formal representation of the OGF of \+ the basic menage problem." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "dsolve(op(1,op(1,ode01)),Y(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %\"YG6#%\"zG*(,**&-%$intG6$**,&!\"\"\"\"\"F'F1F1-%$expG6#*&,&F1F1*$F' \"\"#F1F1F'F0F1,&F'F1F1F1F0F'F0F'F1F'F1F1*&F'F1%$_C1GF1F1F+F1F;F1F1F'F 0-F36#,$F5F0F1" }}}{PARA 0 "" 0 "" {TEXT -1 50 "This guess can be vali dated by direct integration:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "P01:=int(subs([z=-z,u=-t],g01)*exp(-t),t=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$P01G,$**-%#EiG6$\"\"\",$*&,(F*F*%\"zG\"\" #*$F.F/F*F*F.!\"\"F1F*,&F.F*F*F*F*F.F1-%$expG6#,$*&F2F/F.F1F1F*F1" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXUT 307 15 "Hypergeo metrics" }{TEXT -1 61 ". Alternatively any expression with the exponen tial integral " }{XPPEDIT 18 0 "Ei" "I#EiG6\"" }{TEXT -1 106 " can be expressed in terms of a divergent hypergeometrics (that is also the O GF of permutations). One has" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "y*exp(y)*Ei(1,y)=asympt(y*exp(y)*Ei(1,y),y,10);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*(%\"yG\"\"\"-%$expG6#F%F&-%#EiG6$F&F%F&,6F&F&*$F%! \"\"F/*$F%!\"#\"\"#*$F%!\"$!\"'*$F%!\"%\"#C*$F%!\"&!$?\"*$F%F5\"$?(*$F %!\"(!%S]*$F%!\")\"&?.%-%\"OG6#*$F%!\"*F&" }}}{PARA 0 "" 0 "" {TEXT -1 30 "that is also expressible as a " }{XPPEDIT 18 0 "``[" "&%!G6\"" }{XPPEDIT 18 0 "``[2]*F[0]" "*&&%!G6#\"\"#\"\"\"&%\"FG6#\"\"!F'" } {TEXT -1 16 "-hypergeometric:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "hypergeom([1,1],[],z)=series(hypergeom([1,1],[],z),z=0,12);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$\"\"\"F(7\"%\"zG+=F* F(\"\"!F(\"\"\"\"\"#\"\"#\"\"'\"\"$\"#C\"\"%\"$?\"\"\"&\"$?(\"\"'\"%S] \"\"(\"&?.%\"\")V\"'!)GO\"\"*\"(+)GO\"#5\")+o\"*R\"#6-%\"OG6#F(\"#7" }} }{PARA 0 "" 0 "" {TEXT -1 47 "Thus the menage OGF has an hypergeometri c form:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "hg_menage:=1/(1+z )*hypergeom([1,1],[],z/(1+z)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% *hg_menageG*&,&%\"zG\"\"\"F(F(!\"\"-%*hypergeomG6%7$F(F(7\"*&F'F(F&!\" #F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "series(hg_menage,z=0 ,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+5%\"zG\"\"\"\"\"!F%\"\"$\"\" $\"\"%\"#;\"\"&\"#'*\"\"'\"$v'\"\"(\"%8a\"\")\"&+)[\"\"*-%\"OG6#F%\"#5 " }}}{PARA 0 "" 0 "" {TEXT -1 142 "This gives a single alternating sum for the coefficients that is the counterpart of the one of [Comtet, 1 974] for the circular menage problem." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 112 " This process gives rise to a genera l procedure for conversion from exponential integral to hypergeometric form:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "convert_hypergeom :=proc(e) subs(Ei=proc(a,b) if a<>1 then ERWROR(`unable to convert`) el se exp(-b)/b*hypergeom([1,1],[],-1/b) fi end,e); simplify(\"); end;" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#>%2convert_hypergeomG:6#%\"eG6\"F(F(C $-%%subsG6$/%#EiG:6$%\"aG%\"bGF(F(F(@%09$\"\"\"-%&ERRORG6#%2unable~to~ convertG*(-%$expG6#,$9%!\"\"F6F@FA-%*hypergeomG6%7$F6F67\",$*$F@FAFAF6 F(F(F5-%)simplifyG6#%\"\"GF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "convert_hypergeom(P01);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&- %*hypergeomG6%7$\"\"\"F(7\"*&,(F(F(%\"zG\"\"#*$F,F-F(!\"\"F,F(F(,&F,F( F(F(F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 276 11 "Asymptotics" }{TEXT -1 25 ". Experimentally, we have" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "seq(op(i,l01)/(i-1)!*1.0,i=1..16); " }}{PARA 12 "" 1 "" {XPPMATH 20 "62$\"#5!\"\"\"\"!F&$\"+nmmm;!#5$\"++ ++]7F)$\"+LLLL8F)F,$\"+9dGR8F)$\"+@*4DM\"F)$\"+yrzW8F)$\"+r&GkM\"F)$\" +MjjZ8F)$\"+sWb[8F)$\"+y%o#\\8F)$\"+GX$)\\8F)$\"+n2H]8F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "exp(-2)=exp(-2.);" }}{PARA 11 "" 1 "" {XPXPMATH 20 "6#/-%$expG6#!\"#$\"+KGN`8!#5" }}}{PARA 0 "" 0 "" {TEXT -1 108 "Now, two values are forbidden at each position. Empirica lly, the number of menage numbers is seen to satisfy" }}{PARA 267 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "P[n]=n!*exp(-2)*(1+o(1))" "/&%\"PG 6#%\"nG*(-%*factorialG6#F&\"\"\"-%$expG6#,$\"\"#!\"\"F+,&\"\"\"F+-%\"o G6#\"\"\"F+F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 165 "This is to be compared to the asymptotics of derangements. The asymptotic est imate can be established directly from the alternating sum expressio n of coefficients." }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 27 "A simplifie d menage problem" }}{PARA 0 "" 0 "" {TEXT -1 9 "The case " }{XPPEDIT 18 0 "Omega=\{1\}" "/%&OmegaG<#\"\"\"" }{TEXT -1 49 " is in fact a var iant of the derangement problem." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "G1:=build_grammar(\{1\},1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "g1:=subs(gfsolve(op(2,G1),op(3,G1),z,[[u,marked]]),s[ \{\}](z,u));" }}{PARA 11 "" 1 "" {XPPMATH 20Y "6#>%#g1G*&,&!\"\"\"\"\"% \"zGF(F(,(F'F(*&F)F(%\"uGF(F(F)F(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "series(subs([z=-z,u=-t],g1),z=0,11):map(proc(x) int(x *exp(-t),t=0..infinity) end,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+; %\"zG\"\"\"\"\"!F%\"\"\"F%\"\"#\"\"$\"\"$\"#6\"\"%\"#`\"\"&\"$4$\"\"' \"%>@\"\"(\"&(o;\"\")\"'H$[\"\"\"*\"(d%o9\"#5-%\"OG6#F%\"#6" }}}{PARA 0 "" 0 "" {TEXT -1 25 " The sequence appears as " }{TEXT 285 5 "M2905 " }{TEXT -1 33 " in [EIS] where the EGF is given:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "exp(-z)/(1-z)^2=series(exp(-z)/(1-z)^2,z=0,8) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$expG6#,$%\"zG!\"\"\"\"\",&F +F+F)F*!\"#+5F)F+\"\"!F+\"\"\"#\"\"$\"\"#\"\"##\"#6\"\"'\"\"$#\"#`\"#C \"\"%#\"$.\"\"#S\"\"&#\"%>@\"$?(\"\"'#\"&(o;\"%S]\"\"(-%\"OG6#F+\"\") " }}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 16 "2-Menage numbers" }}{PARA 0 "" 0 "" {TEXT -1 53 "This is a generalization of the menage problem \+ where " }{XPPEDIT 18 0 "Omega=\{0,1,2\}" "/%&OmegaG<%\"\"!\"\"\"\"\"# " }{TEZXT -1 78 ": In order to avoid family fights, couples are now at \+ distance at least 3 (!)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " G012:=build_grammar(\{0,1,2\},2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#> %%G012G7%&%\"sG6#<\"<+/%'markedG%(EpsilonG/&%\"aG6#\"\"!%%AtomG/%)dont careG-%%ProdG6$F3F,/&F06#\"\"#F3/&F'6#<#F2-%&UnionG6%-F76$F5F&-F76$&F0 6#\"\"\"F>-F76$F:&F'6#<#FJ/F&-FB6'F-FD-F76$F/F&FFFK/&F'6#<$F2FJ-FB6$-F 76$F5F>-F76$F:FV/FM-FB6%Fen-F76$F/F>Fgn/FHF3%+unlabelledG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "gfsolve(op(2,G012),op(3,G012),z,[[u ,marked]]); g012:=subs(\",s[\{\}](z,u));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<+/-&%\"aG6#\"\"!6$%\"zG%\"uGF+/-%'markedGF*F,/-&F'6#\"\"#F*F+/- &%\"sG6#<#F)F**(F+\"\"\"F,F<,,*$F+\"\"$F<*&F+F4F,FF%%g012G,$*(,,*$%\"zG\"\"$\["\"\"*&F)\"\"#%\"uGF+!\"\"F)!\"#*&F)F +F.F+F/F+F+F/,*F+F+F)F0F,F/F(F+F+,&F/F+F)F+F/F/" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 96 "series(subs([z=-z,u=-t],g012),z=0,13):P012_ser :=map(proc(x) int(x*exp(-t),t=0..infinity) end,\");" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%)P012_serG+9%\"zG\"\"\"\"\"!F'\"\"%\"\"&\"\"&\"#L\" \"'\"$O#\"\"(\"%=>\"\")\"&Su\"\"\"*\"'\\c<\"#5\"(r@%>\"#6\")`jRB\"#7-% \"OG6#F'\"#8" }}}{PARA 0 "" 0 "" {TEXT -1 42 "Only the beginning of th is sequence (till " }{XPPEDIT 18 0 "17440" "\"&Su\"" }{TEXT -1 40 " in cluded) appears in [EIS] as sequence " }{TEXT 269 7 "M3970. " }{TEXT -1 92 "The computation above makes it possible to extend the values of [EIS] to an arbitrary order." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 278 15 "Holonomic forms" }{TEXT -1 27 ". Since t he denominator of " }{XPPEDIT 18 0 "g012" "I%g012G6\"" }{TEXT -1 17 " \+ has degree 1 in " }{XPPEDIT 18 0 "u" "I\"uG6\"" }{TEXT -1 99 ", the La place-like integral applied to it must be expressible in term\s of the \+ exponential integral " }{XPPEDIT 18 0 "Ei(1,z)" "-%#EiG6$\"\"\"%\"zG" }{TEXT -1 65 ". This, in turn is equivalent to a divergent hypergeomet ric form." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "convert(subs([z =-z,u=-t],g012),parfrac,u);h012:=int(\"*exp(-t),t=0..infinity);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,,*$%\"zG\"\"$!\"\"*&F'\"\"#%\"tG \"\"\"F-F'F+*&F'F-F,F-F)F-F-F),*F-F-F'F+F*F-F&F)F-,&F)F-F'F)F)F)" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%%h012G*,,,*&-%#EiG6$\"\"\",$*(,(*$% \"zG\"\"$F+F0!\"#!\"\"F+F+F0F3,&F3F+F0F+F3F3F+-%$expG6#*(,&F+F+F0\"\"# F+F0F3F4F3F+F3*(F(F+F0F+F5F+F2*&F0F1-F66#*&F0F:F4F3F+F+*&F0F:F=F+F3*(F (F+F0F1F5F+F+F+F0F3F4F2,&F0F+F+F+F3-F66#,$F?F3F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "hh012:=convert_hypergeom(h012);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&hh012G*(,&%\"zG\"\"\"-%*hypergeomG6%7$F(F(7\" *(,(*$F'\"\"$F(F'!\"#!\"\"F(F3F'F(,&F3F(F'F(F(F3F(,&F'F(F(F(F3F4F3" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "series(\",z=0,12);" }} {PARA 11 "" 1 "" {XPPMATH ]20 "6#+7%\"zG\"\"\"\"\"!F%\"\"%\"\"&\"\"&\"# L\"\"'\"$O#\"\"(\"%=>\"\")\"&Su\"\"\"*\"'\\c<\"#5\"(r@%>\"#6-%\"OG6#F% \"#7" }}}{PARA 0 "" 0 "" {TEXT -1 40 "One can then get the holonomic f orms by " }{XPPEDIT 18 0 "gfun" "I%gfunG6\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "holexprtodiffeq(hh012,y(z)): subs(_ C[0]=1,\");" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,4*&,2*$%\"zG\"\"(!\"#* $F'\"\")!\"\"*$F'\"\"%!\"%*$F'\"\"*\"\"\"*$F'\"\"'\"\"#*$F'F5F.F,F2F'F 2F2-%\"yG6#F'F2F2*&,0F&F)F*F)F0F2F3\"\"&F-F/*$F'F " 0 "" {MPLTEXT 1 0 25 "diffeqtorec(\",y(z),u(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<+/-%\"uG6#\"\"!\"\"\"/-F&6#F)F(/-F&6#\"\"#F(/-F&6#\"\" $F(,4-F&6#%\"nGF)*&,&F8F)F)F)F)-F&6#F:F)F)*&,&!\"%F)F8!\"\"F)-F&6#,&F8 F)F0F)F)F)*&,&F8!\"$!\"*F)F)-F&6#,&F8F)F4F)F)F)*&,&F8F0\"\")F)F)-F&6#, &F8F)\"\"%F)F)F)*&,&\"#6F)F8F4F)-F&6#,&F8F)\"\"&F)F)F)*&,&!#5F)F8F@F)- F&6#,&F8F)\"\"'F)F)F)*&,&F8F@!\"(F)F)-F&6#,&F8F)\"\"(F)F)F)^-F&6#,&F8F) FMF)F)/-F&6#F`o\"$O#/-F&6#FQF)/-F&6#FXFX/-F&6#Fin\"#L" }}}{PARA 0 "" 0 "" {TEXT -1 55 " This gives a fast procedure for computing the numbe rs." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "s012:=rectoproc(\",u( n),remember);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%s012G:6#%\"nG6\"6# %)rememberGE\\s)\"\"'\"#L\"\"!\"\"\"\"\"#F.F/F.\"\"%F/\"\"&F2\"\"(\"$O #\"\"$F.,4-9!6#,&9$F/!\")F/!\"\"-F86#,&F;F/!\"(F/F3-F86#,&F;F/!\"'F/! \"%-F86#,&F;F/!\"&F/!#:-F86#,&F;F/FFF/\"\")-F86#,&F;F/!\"$F/\"#8-F86#, &F;F/!\"#F/F0-F86#,&F;F/F=F/F=*&,0F>F/FBF=FGFSFLF0FPF5FUF=FYF=F/F;F/F= F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "seq(s012(j),j=0..30 );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6A\"\"\"\"\"!F$F$F#\"\"&\"#L\"$O# \"%=>\"&Su\"\"'\\c<\"(r@%>\")`jRB\"*gf00$\"+k:s!G%\",))y3IV'\".`f(=$3. \"\"/HPb[ea<\"00rJs3:;$\"13Vg*4w?,'\"3a&yx$H3I.7\"47,<'))>e_GD\"5$*4y^ u;bylb\"7.LT!Q0YSO2G\"\"8H88n:jDSR]2$\"9'*o)3Y)*\\y%pR!p(\";7vm\"\\3)e -Uo9+?\"<7dTHkNg!p&=T>S&\">4q)G%>;)\\5,2(HH^\"\"?T*=M!e$*Q]n5)>+&)Q%\" A@zX]z/X$*>S_]6#o_J\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 279 11 "Asymptotics" }{TEXT -1 151 ". This sequence obeys a g eneral asymptotic pattern that was alluded to before. Verification to \+ high orders is easy thanks to the holonomic recurrences." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "seq(1.0*s012(m)/m!,m=0..50);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6U$\"#5!\"\"\"\"!F&F&$\"+nmmmT!#6F'$\"+L LL$e%F)$\"+$oRDo%F)$\"+WW%pv%F)$\"+tk*f![F)$\"+kbTS[F)$\"+Syal[F)$\"+g (*R%)[F)$\"+Mq!*)*[F)$\"+/oI5\\F)$\"+]rU>\\F)$\"+&ePo#\\F)$\"+?.%H$\\F )$\"+3f-Q\\F)$\"+p%3B%\\F)$\"+0'[f%\\F)$\"+&oo!\\\\F)$\"+oKw^\\F)$\"+d j5a\\F)$\"+Ql:c\\F)$\"+z1'z&\\F)$\"+Nmbf\\F)$\"+^_(4'\\F)$\"+i=Ci\\F)$ \"+PuPj\\F)$\"+Z%*Rk\\F)$\"+ZDKl\\F)$\"+-\"fh'\\F)$\"+-'>p'\\F)$\"+'*H hn\\F)$\"+apCo\\F)$\"+z!G)o\\F)$\"+%3i$p\\F)$\"+HR&)p\\F)$\"+QzIq\\F)$ \"+'*ysq\\F)$\"+Br6r\\F)$\"+[&y9(\\F)$\"+cZ\"=(\\F)$\"+S!G@(\\F)$\"+Q/ Us\\F)$\"+mPps\\F)$\"+Z'\\H(\\F)$\"+N&*=t\\F)$\"+NZTt\\F)$\"+Bkit\\F) " }}}{PARA 0 "" 0 "" {TEXT -1 123 "Such a numeri`cal computation suppor ts the assumption that asymptotically, the number of such permutations should grows like" }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "p[n]=n!*exp(-3) *(1+o(1))" "/&%\"pG6#%\"nG*(-%*factorialG6#F&\"\"\"-%$expG6#,$\"\"$!\" \"F+,&\"\"\"F+-%\"oG6#\"\"\"F+F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "In effect, one has:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "exp(-3)=exp(-3.0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG 6#!\"$$\"+Poqy\\!#6" }}}{PARA 0 "" 0 "" {TEXT -1 61 "(See the conclusi on section for a proof of this observation.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 263 "" 0 "" {TEXT 286 28 "Variants of 2-menage numbers" }}{PARA 0 "" 0 "" {TEXT -1 9 "The case " }{XPPEDIT 18 0 "Ome ga=\{1,2\}" "/%&OmegaG<$\"\"\"\"\"#" }{TEXT -1 42 " is close to the cl assical menage problem." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "G 12:=build_grammar(\{1,2\},2):gfsolve(op(2,G12),op(3,G12),z,[[u,marked] ]): g12:=subs(\",s[\{\}](z,u));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ g12G,a$*&,*\"\"\"F(*&%\"zGF(%\"uGF(!\"\"F*!\"#*$F*\"\"#F(F,,*F,F(F*F/F. F,*&F*F/F+F(F(F(F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "serie s(subs([z=-z,u=-t],g12),z=0,11):map(proc(x) int(x*exp(-t),t=0..infinit y) end,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+;%\"zG\"\"\"\"\"!F%\" \"\"F%\"\"#F%\"\"$\"\"&\"\"%\"#B\"\"&\"$J\"\"\"'\"$$))\"\"(\"%fo\"\") \"&,.'\"\"*\"'0;f\"#5-%\"OG6#F%\"#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 16 "3-Menage numbers" }}{PARA 0 " " 0 "" {TEXT -1 32 "The basic version is defined by " }{XPPEDIT 18 0 " Omega=\{0,1,2,3\}" "/%&OmegaG<&\"\"!\"\"\"\"\"#\"\"$" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "G0123:=build_grammar(\{0,1 ,2,3\},3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "gfsolve(op(2, G0123),op(3,G0123),z,[[u,marked]]): g0123:=subs(\",s[\{\}](z,u));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&g0123G,$*&,>*$%\"zG\"\"(\"\"\"*&F) \"\"'%\"uGF+!\"\"*$F)F-F+*&F)\"\"&F.F+F/*$F)F2F+*$F)\"\"%!\"$*&F)F5F.F +F6*$F)\"\"$F/*&F)F9F.F+F+*&F)F9F.\b"\"#F+*&F)FF5F4F " 0 "" {MPLTEXT 1 0 87 "serie s(subs([z=-z,u=-t],g0123),z=0,11):map(proc(x) int(x*exp(-t),t=0..infin ity) end,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"zG\"\"\"\"\"!F% \"\"&\"\"*\"\"'\"#r\"\"(\"$J'\"\")\"%3f\"\"*\"&U2'\"#5-%\"OG6#F%\"#6" }}}{PARA 0 "" 0 "" {TEXT -1 42 "This sequence is not to be found in [E IS]." }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 32 "Variants of the 3-Menage \+ problem" }}{PARA 0 "" 0 "" {TEXT -1 53 "We just offer here a few rando m samples of sequences." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "First, the case " }{XPPEDIT 18 0 "Omega=\{0,3\}" "/% &OmegaG<$\"\"!\"\"$" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "G03:=build_grammar(\{0,3\},3):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 69 "gfsolve(op(2,G03),op(3,G03),z,[[u,marked]]): g 03:=subs(\",s[\{\}](z,u));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%c$g03G, $*&,:*$%\"zG\"\"(\"\"\"*&F)\"\"'%\"uGF+!\"\"*$F)F-F/*&F)\"\"&F.F+F/*$F )\"\"%!\"#*&F)F4F.F+F/*$F)\"\"$\"\"#*&F)F8F.F9F+*&F)F8F.F+F9*&F)F9F.F+ F+F)F+F/F+F+,@F+F+*&F)F+F.F+F/F;F/F7F5*$F)F9F+*&F)F4F.F9F+F6F4F3F4F)F5 *$F)F2F5F1F/F0F+*&F)F*F.F+F/F(F5*$F)\"\")F+F/F/" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 85 "series(subs([z=-z,u=-t],g03),z=0,11):map(proc( x) int(x*exp(-t),t=0..infinity) end,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+9%\"zG\"\"\"\"\"!F%\"\"#\"\"#\"\"$\"\"'\"\"%\"#E\"\"&\"$U\"\"\" '\"$L*\"\"(\"%Or\"\")\"&T@'\"\"*\"'Ndg\"#5-%\"OG6#F%\"#6" }}}{PARA 0 " " 0 "" {TEXT -1 15 "Next, the case " }{XPPEDIT 18 0 "Omega=\{0,1,3\}" "/%&OmegaG<%\"\"!\"\"\"\"\"$" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "G013:=build_grammar(\{0,1,3\},3):gfsolve(op(2,\") ,op(3,\"),z,[[u,marked]]): g013:=subs(\",s[\{\}](z,u));series(subs([z= -z,u=-t],g013),z=0,11):map(proc(x) int(x*exp(-t),t=0..infinity) end,\" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%g013G,$*&,6!\"\"\"\"\"%\"zG\" \"#*&F*\"\"$%\"uGdF)F+*&F*F-F.F+F)*$F*\"\"%!\"#*&F*F+F.F)F)*&F*F1F.F)F2 *$F*\"\"(F)*&F*\"\"'F.F)F(*&F*\"\"&F.F)F(F),:*$F*F+F+*&F*F)F.F)F(F)F)* &F*F1F.F+F)F4F1F*!\"$F9F2*&F*F6F.F)F(F5F(*$F*\"\")F)*$F*F:F2F0F+F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+7%\"zG\"\"\"\"\"!F%\"\"$\"\"#\"\"% \"\")\"\"&\"#U\"\"'\"$)G\"\"(\"%`A\"\")\"&6+#\"\"*\"'M\")>\"#5-%\"OG6# F%\"#6" }}}{PARA 0 "" 0 "" {TEXT -1 13 "And the case " }{XPPEDIT 18 0 "Omega=\{0,2,3\}" "/%&OmegaG<%\"\"!\"\"#\"\"$" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "G023:=build_grammar(\{0,2,3 \},3):gfsolve(op(2,\"),op(3,\"),z,[[u,marked]]): g023:=subs(\",s[\{\}] (z,u));series(subs([z=-z,u=-t],g023),z=0,11):map(proc(x) int(x*exp(-t) ,t=0..infinity) end,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%g023G,$ *&,6*$%\"zG\"\"(\"\"\"*&F)\"\"'%\"uGF+!\"\"*&F)\"\"&F.F+F/*&F)\"\"%F.F +!\"#*$F)F3F4*&F)\"\"$F.\"\"#F+*&F)F7F.F+F+*&F)F8F.F+F8F)F8F/F+F+,:*$F )F8F8*&F)F+F.F+F/F+F+*&F)F3F.F8F+F2F3F)!\"$F0F4*&F)F*F.F+F/F(F/*$F)\" \")F+*$F)F1F4F5F8F/F/" }}{PARA 11 "" 1 "" {XPPMATHe 20 "6#+9%\"zG\"\"\" \"\"!F%\"\"#F%\"\"$\"\"#\"\"%\"#5\"\"&\"#]\"\"'\"$G$\"\"(\"%JD\"\")\"& ;A#\"\"*\"'Ay@\"#5-%\"OG6#F%\"#6" }}}{PARA 0 "" 0 "" {TEXT -1 18 "Fina lly, the case " }{XPPEDIT 18 0 "Omega=\{1,3\}" "/%&OmegaG<$\"\"\"\"\"$ " }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "G13:=bu ild_grammar(\{1,3\},3):gfsolve(op(2,\"),op(3,\"),z,[[u,marked]]): g13: =subs(\",s[\{\}](z,u));series(subs([z=-z,u=-t],g13),z=0,11):map(proc(x ) int(x*exp(-t),t=0..infinity) end,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$g13G,$*&,0*&%\"zG\"\"$%\"uG\"\"#!\"\"*&F)\"\"%F+\"\"\"F0F0F0* &F)F,F+F0F-F)!\"#*$F)F*F,*$F)F/F-F0,.F-F0F)F,*&F)F0F+F0F0*&F)F*F+F0F-F 3F2F4F0F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+;%\"zG\"\"\"\"\"!F%\" \"\"F%\"\"#\"\"$\"\"$\"\"(\"\"%\"#J\"\"&\"$h\"\"\"'\"%X5\"\"(\"%.z\"\" )\"&0#o\"\"*\"'4)f'\"#5-%\"OG6#F%\"#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}}{SECT 0 {PARA 268 "" 0 "" {TEXT -1 16 "4-Menage numbers" }} {PARA 0 "" 0 "" {TEXT -1 188 " This and the next example serve to test the system fon models of a fairly large size. The rational GF for the \+ 4-menage proble is of degree 16, and the computations only take a few \+ seconds." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "G01234:=build_gr ammar(\{0,1,2,3,4\},4):gfsolve(op(2,G01234),op(3,G01234),z,[[u,marked] ]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "g01234:=subs(\",s[\{ \}](z,u));\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'g01234G,$*(,\\p!\" \"\"\"\"%\"zG\"\"%*&F*\"\"&%\"uG\"\"#!\")*$F*F/!\"#*&F*\"\"$F.F)F)*&F* F4F.F/F/*$F*F+F2*&F*F/F.F)F4*&F*F+F.F)!\"&*$F*\"#5!#6*&F*\"\"*F.F/F-*& F*F;F.F)F/*&F*F+F.F/F4*&F*F+F.F4F)*&F*\"\"'F.F/!\"**$F*\"#:F)*&F*\"#8F .F)F(*&F*\"#6F.F/!\"$*&F*FJF.F)F2*&F*\"#7F.F)F)*&F*\"#9F.F)F(*$F*FJF0* $F*FNF2*$F*FPF/*&F*F;F.F4F)*&F*\"\")F.F4F(*&F*F>F.F4F)*$F*F4!\"%*$F*F- F9*$F*FCFN*$F*\"\"(FN*&F*FCF.F4F(*&F*FgnF.F/F2*&F*F>F.F)FP*&F*FgnF.F4F (*&F*FCF.F)F/*&F*F-F.F)!#A*&F*FVF.F)FV*&F*FgnF.F)FN*$F*FVF+F),HF:F)Fjn F(F_oF/FaoF2FinF(F\\oF9FenFYF,F/F]oFCFZF/F8F-F6F+F5F(F3FYFXF2F1F+F*FY* &F*F)F.F)F(F)F)F(,0FenF)FZF)F8F(F7Fg(F1F2F*F(F)F)F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "series(subs([z=-z,u=-t],g01234),z=0,11):m ap(proc(x) int(x*exp(-t),t=0..infinity) end,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"zG\"\"\"\"\"!F%\"\"'\"#<\"\"(\"$f\"\"\")\"%`<\"\"* \"&$>>\"#5-%\"OG6#F%\"#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "convert(g01234,parfrac,u);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(*(,(* $%\"zG\"\"%\"\"\"*$F'\"\"$F)!\"\"F)F)F'F,,&F&F)F,F)F,F,*,,F*$F'\"#5F)* &F'\"\"*%\"uGF)F,*&F'\"\")F4F)F+*$F'F6!\"#*&F'\"\"(F4F)F)*$F'\"\"'!\"% *&F'FFFF;F=*&F'FAF4FBFBFCF " 0 "" {MPLTEXT 1 0 93 "G012345:=build_grammar(\{0,1,2,3,4,5\},5): gfsol ve(op(2,G012345),op(3,G012345),z,[[u,marked]]):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "g012345:=subs(\",s[\{\}](z,u));\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(g012345G,$*&,h\\l*&%\"zG\"#6%\"uG\"\"%\"#C*& F)\"#7F+F,F/*&F)\"\")F+\"\"#\"#>*$F)\"#<\"$%Q*$F)\"#;\"$/#*&F)\"#9F+F, \"#J*&F)\"#=F+\"\"$!#U*&F)F3F+F2\"#&**$F)F>\"$i#*&F)F8F+F?!$=#*&F)\"#: F+F?!#o*&F)F8F+\"\"\"!$=$*&F)F3F+FK\"#W*&F)FHF+F,F**$F)\"#8!$i#!\"\"FK *&F)F;F+F2F5*&F)FQF+F2!#x*&F)FQF+F?\"\"&*&F)\"#5F+F,!#<*&F)F*F+F?\"$t \"*&F)F/F+F?\"#v*&F)F/F+F2F>*&F)F;F+F?\"#RF)FX*&F)F8F+F2!$L#*&F)FHF+F2 !$_\"*$F)F3\"$Y\"*&F)F5F+F2\"$0\"*&F)FXF+F2!\"&*&F)FQF+F,F1*&F)F>F+F2 \"$C\"*&F)F>F+FK\"#im*&F)FHF+FK!$%R*&F)F5F+F?!#k*&F)F5F+FK\"#[*&F)F8F+F ,!#b*&F)FZF+F2\"$\\#*$F)F2!\"%*&F)F?F+F2F?*&F)FXF+F?\"\"'*$F)F,!\"#*&F )F3F+F?!#D*&F)\"#BF+F2F<*&F)F-F+F2\"#A*&F)\"#@F+F2\"$A\"*&F)F`qF+FK!#Q *&F)FbqF+F2\"#N*&F)F-F+FK!#K*&F)\"#?F+FK\"$1\"*&F)FbqF+FK!#E*&F)F]rF+F ?FS*&F)FdqF+F?\"#H*&F)F]rF+F,!\"(*&F)F]rF+F2\"$@\"*&F)FdqF+FK\"##)*&F) F3F+F,!#8*$F)F]r\"#!)*$F)Fdq!#S*&F)F`qF+F,F?*&F)\"\"(F+F,F\\q*&F)F1F+F ,!\")*&F)F2F+FKF,*&F)F>F+FX!\"$*&F)F-F+F?Fer*&F)\"#EF+F?Fgs*&F)\"#DF+F ,FK*$F)F-!#X*$F)F`q!#s*$F)Fbq!$7\"*&F)F,F+FKFds*$F)FZF[r*&F)\"#IF+FKFS *&F)F/F+FXF2*&F)F;F+FXF1*&F)FZF+FXF\\q*&F)FHF+FjpFK*&F)F]rF+FXF\\q*&F) \"\"*F+F2F[p*&F)FZF+FK\"$u#*$F)F]uF1*&F)F,F+F2FX*&F)F,F+F?F2*&F)FjpF+F 2!##**&F)FHF+FXFjp*&F)F]uF+F,!#;*&F)FQF+FXF2*&F)F*F+FXF2*&F)F5F+F,!#?* &F)F8F+FX!\"'*$F)FHFgp*&F)FQF+FK!$K$*&F)F*F+F2\"$P%*&F)F*F+FK\"$Y#*&F) F/F+FK!$'H*&F)F;F+FK!$Y#*$F)F*F`o*$F)F/!$q$*$F)F;!$/\"*&F)FZF+F?Fbs*&F )F1F+F?!#V*&F)F]uF+F?!#\\*&F)F>F+F,!#>*&F)F5F+FXF\\q*$F)F?F]v*$F)FXFgp *$F)Fjp!\"**$F)Fbs\"#x*&F)FXF+F,FK*&F)FjpF+F?j!#9*&F)FbsF+F2!#j*&F)F]uF +FK\"$e\"*&F)FbsF+F?!#P*&F)FjpF+FK!$0\"*&F)FXF+FK!#J*&F)F1F+FK\"$u\"*& F)FbsF+FK\"$3\"*$F)F1\"#k*&F)F]uF+FXFS*&F+FXF)F3F\\q*&F)F;F+FjpFK*$F) \"#FFZ*&F+F2F)FjsF?*&F+F2F)F\\tFQ*&F+F?F)F`qFer*&F+F?F)FbqFgs*&F+F?F)F \\tFgs*&F+F,F)FdqFbs*&F+F,F)F-FK*&F+F,F)FbqF2*&F+FKF)FcyF]v*&F+FKF)Fjs !#A*&F+FKF)F\\t!#G*&F+FKF)\"#GF\\q*&F+FKF)FcrFS*&F+F2F)FcyFK*$F)FjsFgp *$F)F\\tFhx*$F)FbzFjp*$F)FcrF,*$F)FftF?*$F)FFfqF1Fhq\"$+\"FjqFdsF\\r!#[F_r\"$?\"FarF_[lFbrFhxFdrFgpFfrF]\\lFhrFap Fjr!#7F\\sF[pF^sFf\\l*&F)FjpF+F,FKFcsF\\qF]t!#FF_tF_sFatFf[lFctF,FdtF` [lF\\uF\\qF^u!#cF`u\"#cFauFKFcuF[pFeuF2FhuF2FiuF\\qFju!#_F\\vF/F^v!$+ \"F_vF@Fav!#yFcv\"$7\"Fev\"$+'Fgv!$o\"FivFf\\lFjv\"$=#F\\w!$e\"F^wF_pF _wkF_[lFawFfoFcwFgsFewF]vFfwF2FgwF2FhwFXFjw!#')F]xF8F_xFbtFaxFf[lFcxFg \\lFexF^yFgxF3FixF[rF[y!$P#F]yFQF`yF\\q*&F+FXF)FdqF\\qFbyF^x*&F)F8F+Fj pFKFdyFZFeyF1*&F+F2F)FbzFKFfyF?Fgy\"#K*&F+F?F)FcyFgsFhyFgpFiyFg\\lFjyF 2F[zFX*&F+F,F)FjsFKF\\zFenF]zFdsF_zFfoFazFgpFczFS*&F+FKF)F " 0 "" {MPLTEXT 1 0 89 "series(subs([z=-z,u=-t],g012345),z=0,13):map(pro c(x) int(x*exp(-t),t=0..infinity) end,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"zG\"\"\"\"\"!F%\"\"(\"#L\"\")\"$r$\"\"*\"%H]\"#5\" &6V'\"#6\"'([f)\"#7-%\"OG6#F%\"#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "denom(g012345):series(\",u,8):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 14 "map(factor,\");" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+1%\"uG*(,&!\"\"\"\"\"%\"zGF(\"\"#,.*$F)\"\"&F(*$F)\"\"%F(*$F)\" \"$F(*$F)F*F(F)F(F'F(F*,6*$F)\"#5F(*$F)\"\"*F(*$F)\"\"(F(*$F)\"\"'F(F, !\"'F.!\"&F0!\"$F2!\"#F)F'F(F(F*\"\"!,$**F)F(F&F(F+F(,R*$F)\"#CF(*$F) \"#AF(*$F)\"#@F/*$F)\"#?\"#<*$F)\"#>F5*$F)\"l#=F/*$F)FLF5*$F)\"#;F'*$F) \"#:!#!**$F)\"#9!#X*$F)\"#8\"#Y*$F)\"#7!#I*$F)\"#6!#GF4\"$J#F6\"$g\"*$ F)\"\")\"#^F8\"#iF:FSF,!$5\"F.!#BF0F?F2F(F)F*F(F(F(F'\"\"\"*&F)F/,TFQF PFR!#5FO\"$+\"FZ!$'QF(F(F)F5FMF;F2\"#mF.!#qFJF5FHF`oFDF(*$F)\"#BF*FFF5 F4!$A\"F6\"#kFT\"#SFjn\"$u\"Fgn\"#IFWFhoF0!$7\"F,F?F:!$[\"F8!#yF_o\"$U &F(\"\"#,$*&F)F-,LFgn\"$4#FJF/F_o!#sF8!$g\"FFF1FZFEF:\"$2\"F6FapF0FEFR \"#JFjn!#'*F'F(F,F`pF)!#;FOF>FQ!#KF2FhnF4!#&*F.F-FTFEFWFEF(F'\"\"$*&F) F;,DF(F(FRF-Fjn!#_F4\"#aF.!\"(F_oFjqF2F?FW!\"%FgnF>F,F\\oFJF(FOF*F8FKF :FSFZ!#7FTF\\rF6FKF(\"\"%,$*&F)F[o,0F_oF(F.F'F2F'F(F(F,F " 0 "" {MPLTEXT 1 0 237 "build_transition:=proc(state,Omega ,initial_state) local x,t; if state=initial_state then t:=Epsilon else t:=NULL fi;s[state]=Union(t,Prod(dontcare,s[MinusOne(state)]),seq(Pro d(a[x],s[MinusOne(state union \{x\})]),x=Omega minus state)) end;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%1build_transitionG:6%%&stateG%&Omega G%.initial_stateG6$%\"xG%\"tG6\"F-C$@%/9$9&>8%%(EpsilonG>F4%%NULLG/&% \"sG6#F1-%&UnionG6%F4-%%ProdG6$%)dontcareG&F:6#-%)MinusOneGF;-%$seqG6$ -F@6$&%\"aG6#8$&F:6#-FF6#-%&unionG6$F1<#FO/FO-%&minusG6$9%F1F-F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "build_transition(\{1,2\},\{0 ,1,2,3\},\{\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"sG6#<$\"\"\"\" \"#-%&UnionG6%-%%ProdG6$%)dontcareG&F%6#<$\"\"!F(-F.6$&%\"aG6#F4F1-F.6 $&F86#\"\"$&F%6#<%F4F(F)" }}}{PARA 0 "" 0 "" {TEXT -1 102 "The followi ng procedure builds the specification that corresponds to a loop that \+ starts and enpds with " }{XPPEDIT 18 0 "initial_state" "I.initial_state G6\"" }{TEXT -1 80 " in the transition graph. It is a simple modificat ion of the previous procedure " }{XPPEDIT 18 0 "build_grammar" "I.buil d_grammarG6\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "build_loop:=proc(Omega,initial_state) [s[\{\}],\{build_alphabet (Omega)\} union map(build_transition,combstruct[allstructs](Subset(\{$ 0..max(op(Omega))-1\})),Omega,initial_state),unlabelled] end;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%+build_loopG:6$%&OmegaG%.initial_sta teG6\"F)F)7%&%\"sG6#<\"-%&unionG6$<#-%/build_alphabetG6#9$-%$mapG6&%1b uild_transitionG-&%+combstructG6#%+allstructsG6#-%'SubsetG6#<#-%\"$G6# ;\"\"!,&-%$maxG6#-%#opGF5\"\"\"!\"\"FPF69%%+unlabelledGF)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "build_loop(\{0\},\{\});build _loop(\{0,1\},\{0\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%&%\"sG6#<\" <&/F$-%&UnionG6%%(EpsilonG-%%ProdG6$%)dontcareGF$-F/6$&%\"aG6#\"\"!F$/ %'markedGF-/F4%%AtomG/F1-F/6$F;F9%+unlabelledG" }}{qPARA 12 "" 1 "" {XPPMATH 20 "6#7%&%\"sG6#<\"<(/%'markedG%(EpsilonG/&%\"aG6#\"\"!%%Atom G/%)dontcareG-%%ProdG6$F1F*/&F%6#<#F0-%&UnionG6%F+-F56$F3F$-F56$&F.6# \"\"\"F8/F$-F<6%F>-F56$F-F$F@/FBF1%+unlabelledG" }}}{PARA 0 "" 0 "" {TEXT -1 110 " We could construct a composite grammar for all loops, i t is however simpler to solve for the GF of each loop." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 187 "gf_loops:=proc(Omega) local is; \n normal(add(subs(gfsolve(op(2,build_loop(Omega,is)),unlabelled,z,[[u,ma rked]]),s[is](z,u)),is=combstruct[allstructs](Subset(\{$0..max(op(Omeg a))-1\})))) end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)gf_loopsG:6#%&O megaG6#%#isG6\"F*-%'normalG6#-%$addG6$-%%subsG6$-%(gfsolveG6&-%#opG6$ \"\"#-%+build_loopG6$9$8$%+unlabelledG%\"zG7#7$%\"uG%'markedG-&%\"sG6# F?6$FAFD/F?-&%+combstructG6#%+allstructsG6#-%'SubsetG6#<#-%\"$G6#;\"\" !,&-%$maxG6#-F86#F>\"\"\"!\"\"F[oF*F*" }}}{PARA 0 "" 0 "" {TEXT -1 134 "The bivariate GF counts configurations with possible exceptions ( it is useful for the ranalysis of configurations with forbidden gaps). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "In th e case where only singletons are allowed, then there is only 1 possibi lity for each size " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "l0:=gf_loops(\{0\},0); subs( u=0,\");series(\",z=0,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#l0G,$ *$,(!\"\"\"\"\"*&%\"zGF)%\"uGF)F)F+F)F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$,&!\"\"\"\"\"%\"zGF'F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+9%\"zG\"\"\"\"\"!F%\"\"\"F%\"\"#F%\"\"$F%\"\"%F%\"\"&F %\"\"'F%\"\"(F%\"\")F%\"\"*-%\"OG6#F%\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 12 "In the case " }{XPPEDIT 18 0 "Omega=\{0,1\}" "/%&OmegaG<$\"\"! \"\"\"" }{TEXT -1 60 ", the first element (either 0 or 1) conditions a ll the rest." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "l01:=gf_loop s(\{0,1\},1);subs(u=0,\");series(\",z=0,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$l01G,$*&,(!\"#\"\"\"*&%\"zGF)%\"uGF)F)F+\"\"#F),*F)F s)F*!\"\"F+F(*$F+F-F)F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&!\"# \"\"\"%\"zG\"\"#F',(F'F'F(F&*$F(F)F'!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+9%\"zG\"\"#\"\"!F%\"\"\"F%\"\"#F%\"\"$F%\"\"%F%\"\"&F% \"\"'F%\"\"(F%\"\")F%\"\"*-%\"OG6#\"\"\"\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 9 "The case " }{XPPEDIT 18 0 "Omega=\{0,1,2\}" "/%&OmegaG<%\" \"!\"\"\"\"\"#" }{TEXT -1 37 " is the first one that is nontrivial." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "l012:=gf_loops(\{0,1,2\},2) ;subs(u=0,\");series(\",z=0,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% %l012G,$*(,.\"\"%\"\"\"%\"zG!\"**&F*F)%\"uGF)!\"$*$F*\"\"#F(*$F*\"\"$F )*&F*F2F-F)F)F),,F1F)*&F*F0F-F)!\"\"F*!\"#F,F6F)F)F6,&F6F)F*F)F6F6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,*\"\"%\"\"\"%\"zG!\"**$F(\"\"#F&* $F(\"\"$F'F',(F,F'F(!\"#F'F'!\"\",&F0F'F(F'F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+9%\"zG\"\"%\"\"!\"\"$\"\"\"\"\"&\"\"#\"\"'\"\"$\"\"*\" \"%\"#8\"\"&\"#?\"\"'\"#J\"\"(\"#\\\"\")\"#y\"\"*-%\"OG6#\"\"\"\"#5" } }}{PARA 0 "" 0 "" {TEXT -1 13 "The sequtence " }{XPPEDIT 18 0 "3,5,6,9, 13" "6'\"\"$\"\"&\"\"'\"\"*\"#8" }{TEXT -1 18 ",etc, is sequence " } {TEXT 287 5 "M2396" }{TEXT -1 5 " of [" }{TEXT 288 3 "EIS" }{TEXT -1 2 "]." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "l0123:=gf_loops(\{0 ,1,2,3\},3);subs(u=0,\");series(\",z=0,12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&l0123G,$*&,8!\")\"\"\"%\"zG\"#G*&F*F)%\"uGF)\"\"(*$F *\"\"#!#C*&F*\"\"$F-F)!\"&*$F*\"\"%F(*&F*\"\"&F-F)\"\"**$F*F8\"#7*&F*F 6F-F)!#;*&F*F6F-F0!\"%*&F*F.F-F)F)F),:F,!\"\"FF)F@FB*$F*\"\")F)FBFB" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$ *&,,!\")\"\"\"%\"zG\"#G*$F(\"\"#!#C*$F(\"\"%F&*$F(\"\"&\"#7F',.F'F'F(! \"%F*F.F-F+F/F3*$F(\"\")F'!\"\"F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+ =%\"zG\"\")\"\"!\"\"%\"\"\"F%\"\"#\"#;\"\"$\"#C\"\"%\"#W\"\"&\"#!)\"\" '\"$W\"\"\"(\"$k#\"\")\"$%[\"\"*\"$)))\"#5\"%K;\"#6-%\"OG6#\"\"\"\"#7 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 290 20 "Gen erating functions" }{TEXT -1 34 ". These are clearly rational. Let " } {XPPuEDIT 18 0 "f(z)=N(z)/D(z)" "/-%\"fG6#%\"zG*&-%\"NG6#F&\"\"\"-%\"DG 6#F&!\"\"" }{TEXT -1 119 " be the GF that arises in each case. Then, a general result concerning generating functions of loops in graphs [Bi ggs, " }{TEXT 308 22 "Algebraic Graph Theory" }{TEXT -1 21 ", 1993] im plies that " }{XPPEDIT 18 0 "f(z)-m" ",&-%\"fG6#%\"zG\"\"\"%\"mG!\"\" " }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "m=2^(d-1)" "/%\"mG)\"\"#,&%\" dG\"\"\"\"\"\"!\"\"" }{TEXT -1 67 " is the dimension of the state spa ce, is a logarithmic derivative," }}{PARA 270 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(z)-m=z*diff(D(z),z)/D(z)" "/,&-%\"fG6#%\"zG\"\"\"%\"m G!\"\"*(F'F(-%%diffG6$-%\"DG6#F'F'F(-F06#F'F*" }}{PARA 0 "" 0 "" {TEXT -1 13 "For instance:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "L0123:=subs(u=0,l0123);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&L012 3G,$*&,,!\")\"\"\"%\"zG\"#G*$F*\"\"#!#C*$F*\"\"%F(*$F*\"\"&\"#7F),.F)F )F*!\"%F,F0F/F-F1F5*$F*\"\")F)!\"\"F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "int((L0123-8)/z,z);" }}{vPARA 11 "" 1 "" {XPPMATH 20 " 6#,$-%#lnG6#,(*$%\"zG\"\"%\"\"\"F+F+F)!\"#F," }}}{PARA 0 "" 0 "" {TEXT -1 38 "This is also true of the bivariate GF:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "int((l0123-8)/z,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%#lnG6#,:*&%\"zG\"\"\"%\"uGF*!\"\"*&F)\"\"%F+F*F.*$F )\"\"#F.*$F)F.F0F*F*F)!\"%*&F)\"\"$F+F*F**$F)\"\"&F2*&F)F6F+F*!\"$*&F) F.F+F0F**&F)\"\"(F+F*F,*$F)\"\")F*F," }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 289 11 "Recurrences" }{TEXT -1 47 ". The GF' s for forced position gaps are always " }{TEXT 317 8 "rational" } {TEXT -1 77 ". Linear recurrences (with constant coefficients) are the n available by gfun." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "seri es(L0123,z=0,12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+=%\"zG\"\")\"\"! \"\"%\"\"\"F%\"\"#\"#;\"\"$\"#C\"\"%\"#W\"\"&\"#!)\"\"'\"$W\"\"\"(\"$k #\"\")\"$%[\"\"*\"$)))\"#5\"%K;\"#6-%\"OG6#\"\"\"\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "holexprtodiffeq(L0123,y(z));" }} {PARA 11 "" 1 "" {wXPPMATH 20 "6#,(*&,(*$%\"zG\"\"%\"\"\"F)F)F'!\"#F)-% \"yG6#F'F)F)F'\"#7!\")F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "recl0123:=diffeqtorec(\",y(z),u(n)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)recl0123G<&,,-%\"uG6#%\"nG!\" \"-F(6#,&F*\"\"\"F/F/F+-F(6#,&F*F/\"\"#F/F+-F(6#,&F*F/\"\"$F/F/\"\"%F/ /-F(6#\"\"!\"\")/-F(6#F/F8/-F(6#F3F=" }}}{PARA 0 "" 0 "" {TEXT -1 95 " High index values can be obtained quickly by the binary powering metho d that is implemented in " }{HYPERLNK 17 "gfun[rectoproc]" 2 "gfun[rec toproc]" "" }{TEXT -1 53 ". The recurrence needs to be made homogeneou s, first." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "op(1,recl0123)- subs(n=n+1,op(1,recl0123));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%\"u G6#%\"nG!\"\"-F%6#,&F'\"\"\"\"\"$F,\"\"#-F%6#,&F'F,\"\"%F,F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "sl0123:=rectoproc(\{\",u(0)= 8,u(1)=4,u(2)=8,u(3)=16\},u(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% 'sl0123G:6#%\"nG6+%\"aG%$resG%\"iG%\"lG%#u0G%#u1G%#ux2G%#u3G%#u4G6\"E\\ s%\"\"#\"\")\"\"$\"#;\"\"!F5\"\"\"\"\"%@%19$\"$c#C(>8(F5>8)F:>8*F5>8+F 7?(8&F:F9,&F=F9!\"\"F9%%trueGC'>8,,&FAFKFGF4>FAFC>FCFE>FEFG>FGFOFPC'>8 $-%&arrayG6%;F9F:Fen7&7&F4F8F8FK7&F9F8F8F87&F8F9F8F87&F8F8F9F8>8%-FY6$ Fen7&F7F5F:F5>8'-%(convertG6%,&F=F9!\"$F9%%baseGF4@$/&Fao6#F9F9>F\\o-F Y6$Fen7&/F4F7/F6F5/F:F:/F9\"#C@%/Fao7#F9&F\\oF[pC$?&FI-%'subsopG6%/F9F 2/-%%nopsG6#FaoF2FaoFLC$>FW-FY6%FenFen72/6$F:F6,**&&FW6$F:F9F9&FW6$F9F 6F9F9*&&FW6$F:F4F9&FW6$F4F6F9F9*&&FWFiqF9&FW6$F6F6F9F9*&&FW6$F:F:F9Ffr F9F9/6$F9F4,**&&FW6$F9F9F9&FWF]sF9F9*&FbsF9&FW6$F4F4F9F9*&F^rF9&FW6$F6 F4F9F9*&&FW6$F9F:F9FarF9F9/F[s,**&FjsF9F\\rF9F9*&&FW6$F4F:F9FarF9F9*&& FW6$F6F:F9FfrF9F9*$FjrF4F9/Fdr,**&&FW6$F4F9F9F^rF9F9*&FdsF9FcrF9F9*&Fc rF9FgrF9F9*&F`tF9FfrF9F9/Fdt,**&&FW6$F6F9F9FjsF9F9*&FgsF9F`tF9F9*&FgrF 9FctF9F9*&FctF9FjrF9F9/Fjt,**&FitF9F`sF9F9*&FdsF9FitF9F9*&FcrF9FauF9F9 *&F`tF9F\\rF9F9/Fat,**&FitF9FjsF9F9*&FdsF9F`tF9F9*&FcrF9FctF9F9*&F`tF9 FjrF9F9/Fhs,**&FauF9FbsF9F9*&FgsF9FdsF9F9*&FgrF9FgsF9F9*&FctF9FarF9F9/ Fes,**&yFbsF9FitF9F9*$FdsF4F9*&FcrF9FgsF9F9F_tF9/F]r,**&F\\rF9F`sF9F9*& FarF9FitF9F9*&FfrF9FauF9F9*&FjrF9F\\rF9F9/F[t,**&F`sF9FjsF9F9*&FbsF9F` tF9F9*&F^rF9FctF9F9*&FjsF9FjrF9F9/F_r,**&F`sF9F^rF9F9*&FbsF9FcrF9F9*&F ^rF9FgrF9F9*&FjsF9FfrF9F9/Fbr,**&F\\rF9FbsF9F9*&FarF9FdsF9F9*&FfrF9Fgs F9F9*&FjrF9FarF9F9/Fas,**$F`sF4F9FjvF9*&F^rF9FauF9F9F^tF9/Fbu,**&FauF9 F`sF9F9*&FgsF9FitF9F9*&FgrF9FauF9F9*&FctF9F\\rF9F9/Fhr,*FhxF9F\\wF9*$F grF4F9FbtF9@$/FIF9>F\\o-FY6$Fen7&/F4,**&FitF9FhpF9F9*&FdsF9&F\\o6#F4F9 F9*&FcrF9&F\\o6#F6F9F9*&F`tF9&F\\o6#F:F9F9/F6,**&FauF9FhpF9F9*&FgsF9F \\zF9F9*&FgrF9F_zF9F9*&FctF9FbzF9F9/F:,**&F\\rF9FhpF9F9*&FarF9F\\zF9F9 *&FfrF9F_zF9F9*&FjrF9FbzF9F9/F9,**&F`sF9FhpF9F9*&FbsF9F\\zF9F9*&F^rF9F _zF9F9*&FjsF9FbzF9F9,**&FfxF9FhpF9F9*&F^sF9F\\zF9F9*&FjwF9F_zF9F9*&Fdw F9FbzF9F9F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "sl0123(100 0);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#\"d[lW:W8d8B_H!>#G;N6Y0$HTQY%\\ V+2d'**4j[%37_$e\"4)37=)oB%=J)\\50t]BUl.q$zlcad*zru>\\!y^u)3lCiIpd0`U# o]B#p!\\))G,<&Q:'3U]wrVLF%f!\\X\\Iz)3;P\\2\"QW7P(**\\!f;/?RX8BA*)" }}} {PARA 0 "" 0 "" {TEXT -1 86 "The complexity of this computation is thu s of O(log(n)) arithmetic operations for the " }{XPPEDIT 18 0 "n" "I\" nG6\"" }{TEXT -1 9 "th term: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "for m in [1000,2000,4000,8000,16000,32000] do st:=time():sl0123( m):print(time()-st); od:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#h!\"$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"##)!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$>\"!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$#=! \"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$>$!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%U7!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 120 "The comput ation of the value of index 32000 takes typically 1 second of computer time though the number has 8470 digits:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "sl0123(32000)*1.0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"++d@67\"%g%)" }}}{PARA 0 "" 0 "" {TEXT -1 106 "(Such fast algorith ms are useful for instance when analyzing patterns in strings, like {in DNA sequences.) " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 23 "Forbidden position gaps" }}{PARA 0 "" 0 "" {TEXT -1 222 "We are dealing here with menage problems in the classical sens e of a circular table. The same integral transformation as in the nonc yclic/nontoroidal case works for obtaining the GF of permutations with excluded patterns, " }{TEXT 292 3 "i.e" }{TEXT -1 22 "., no position \+ gap in " }{XPPEDIT 18 0 "Omega" "I&OmegaG6\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "The case " } {XPPEDIT 18 0 "Omega=\{0\}" "/%&OmegaG<#\"\"!" }{TEXT -1 52 " gives ba ck the derangement numbers (again, this is " }{TEXT 297 5 "M1937" } {TEXT -1 11 " of [EIS])." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 " series(subs([z=-z,u=-t],l0),z=0,13):map(proc(x) int(x*exp(-t),t=0..inf inity) end,\");" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+=%\"zG\"\"\"\"\"!F %\"\"#\"\"#\"\"$\"\"*\"\"%\"#W\"\"&\"$l#\"\"'\"%a=\"\"(\"&L[\"\"\")\"' '\\L\"\"\"*\"(h\\L\"\"#5\")q|Xo9\"#6\"*T[@w\"\"#7-%\"OG6#F%\"#8" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Next come s the case " }{XPPEDIT 18 0 "Omega=\{0,1\}" "/%&OmegaG<$\"\"!\"\"\"" } {TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "series(subs ([z=-z,u=-t],l01),z=0,13):map(proc(x) int(x*exp(-t),t=0..infinity) end ,\");" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+=%\"zG\"\"#\"\"!!\"\"\"\"\" \"\"\"\"\"$F%\"\"%\"#8\"\"&\"#!)\"\"'\"$z&\"\"(\"%QZ\"\")\"&(QV\"\"*\" '#zR%\"#5\"(T2*[\"#6\")Um@f\"#7-%\"OG6#F)\"#8" }}}{PARA 0 "" 0 "" {TEXT -1 113 "This, apart from the first two values (see the next sect ion for possible adjustments) is exactly the sequence of " }{TEXT 294 14 "Menage Numbers" }{TEXT -1 20 " that is listed as " }{TEXT 293 6 " M2062 " }{TEXT -1 9 "in [EIS]." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "int(x*exp(-t)*subs([z=-z,u=-t],l01-2),t=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**%\"xG\"\"\",(*&-%$expG6#!\"#F&-%#EiG6$F& ,$*&,(F&F&%\"zG\"\"#*$F3F4F&F&F3!\"\"F6F&F&*&F3F&-F*6#*&,&F&F&F}5F&F&F3 F6F&F&*(F)F&F-F&F3F4F6F&F3F6-F*6#,$F:F6F&F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "convert_hypergeom(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,*%\"zG\"\"\"*&F&F'-%*hypergeomG6%7$F'F'7\"*&,(F'F' F&\"\"#*$F&F0F'!\"\"F&F'F'F'F)F2F'F'F'%\"xGF',&F&F'F'F'F2F2" }}}{PARA 0 "" 0 "" {TEXT -1 115 "This gives the ordinary GF for the menage prob lem. This GF is closely related to that of the linear menage problem. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The g eneralized menage numbers that correspond to " }{XPPEDIT 18 0 "Omega= \{0,1,2\}" "/%&OmegaG<%\"\"!\"\"\"\"\"#" }{TEXT -1 11 " appear as " } {TEXT 295 5 "M2121" }{TEXT -1 60 " in [EIS] under the name \"discordan t permutations of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 2 " \"." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "series(subs([z=-z,u=- t],l012),z=0,13):map(proc(x) int(x*exp(-t),t=0..infinity) end,\");" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#+=%\"zG\"\"%\"\"!!\"#\"\"\"\"\"\"\"\"# F)\"\"%\"\"#\"\"&\"#?\"\"'\"~$W\"\"\"(\"%l7\"\")\"&s?\"\"\"*\"'ll7\"#5 \"(+^W\"\"#6\")S^(y\"\"#7-%\"OG6#F)\"#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The next sequence called \"hit pol ynomials\" is given as " }{TEXT 296 5 "M2154" }{TEXT -1 46 " in [EIS], where it stops at the value 369321." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "series(subs([z=-z,u=-t],l0123),z=0,13):map(proc(x) in t(x*exp(-t),t=0..infinity) end,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #+=%\"zG\"\")\"\"!!\"$\"\"\"\"\"#\"\"#!\"\"\"\"$\"\"\"\"\"&F)\"\"'\"#J \"\"(\"$k#\"\")\"%$y#\"\"*\"&=3$\"#5\"'@$p$\"#6\"(_fu%\"#7-%\"OG6#F-\" #8" }}}{PARA 0 "" 0 "" {TEXT -1 122 "It is possible to complete the ta ble to arbitrary orders, obtaining exact forms of the GF, recurrences, etc. For instance:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "int(s ubs([z=-z,u=-t],l0123)*exp(-t),t=0..infinity):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "hl0123:=convert_hypergeom(\");" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'hl0123G,$*,%\"zG\"\"#,ho*$*&,0*$F'\"\"'\"\"\"*$ F'\"\"&!\"#*$F'\"\"%\"\"$*$F'F5F4*$F'F(!\"&F'F2F/F/F/,&F/F/F7F/F(#F/F( !\"%*&F+F:F'F/!#7*&F'F4-%*hypergeomG6%7$F/F/7\",$*&,0F7!\"\"F-F/F3F5F6 F4F/F/F " 0 "" {MPLTEXT 1 0 22 "seri es(hl0123,z=0,33);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+[o%\"zG\"\")\" \"!!\"$\"\"\"\"\"#\"\"#!\"\"\"\"$\"\"\"\"\"&F)\"\"'\"#J\"\"(\"$k#\"\") \"%$y#\"\"*\"&=3$\"#5\"'@$p$\"#6\"(_fu%\"#7\")**fFl\"#8\"*EU(y&*\"#9\" ,*=%e^\\\"\"#:\"-?(>S_Z#\"#;\".x$\\?-MV\"#<\"/9lK&R_+)\"#=\"1.#4MD**fb \"\"#>\"2W@a_y5b<$\"#?\"3tP*Q9j'3!z'\"#@\"5YX0fHg,F=:\"#A\"6*40h&e*>`k VN\"#B\"7C_U_SnYa')=')\"#C\"9JDF%o$**Ric2\"=#\"#D\":uEK8NP0q!HIMd\"#E \"<`I[>rV*)H#pO>k:\"#F\"=!GHu'Q$3H!pFjH@W\"#G\"?:&z>O,!eqpsNzU$H\"\"#H \"@5'[M1\"4(pR--WF(=\"R\"#I-%\"OG6#F-\"#J" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "Conclusions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 206 "The combinator ial packages may be used to facilitate the enumeration of permutations with contraints on \"positions\". In essence, a finite-state model is solved that leads to a bivariate generating function " }{XPPEDIT 18 0 "R(z,t)" "-%\"RG6$%\"zG%\"tG" }{TEXT -1 62 " and the GF's of interes t appear to be integral transforms of " }{XPPEDIT 18 0 "R(z,t)" "-%\"R G6$%\"zG%\"tG" }{TEXT -1 108 ". In particular these GF are all holonom ic. Thus, the corresponding counting sequences can be determined in " }{XPPEDIT 18 0 "O(n)" "-%\"OG6#%\"nG" }{TEXT -1 24 " arithmetic operat ions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "If partial fraction decompositions are effected on " }{XPPEDIT 18 0 " R(z,t)" "-%\"RG6$%\"zG%\"tG" }{TEXT -1 111 ", then the GF's appear to \+ be systematically expressible as compositions of the divergent hyperge ometric series " }{XPPEDIT 18 0 "hypergeom([1,1],[],z)" "-%*hypergeomG 6%7$\"\"\"\"\"\"7\"%\"zG" }{TEXT -1 185 " with algebraic functions. Th e simplest of these forms have been made explicit above in some cases, like derangements, menage, and successions. The same method also allo ws for counting " }{XPPEDIT 18 0 "Omega" "I&OmegaG6\"" }{TEXT -1 15 "- exceptions in " }{TEXT 283 4 "all " }{TEXT -1 190 "permutations by mea ns of bivariate generating functions. A typical problem in this range \+ is the statistics of the number of \"conflicts\" in a random assignmen t of seats in the menage problem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 205 "The fast algorithms make it possible to \+ perform numerical observations and conjecture various asymptotic patte rns. One of these, which is confirmed in the particular cases treated \+ above is [see, I. Vardi, " }{TEXT 284 42 "Computational Recreations wi th Mathematica" }{TEXT -1 32 ", Addison-Wesley, 1991; p. 123]:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 309 8 "Property " }{TEXT -1 2 ". " }{TEXT 280 75 "The fraction of permutations with g aps that do not belong to a finite set " }{XPPEDIT 281 0 "Omega" "I&Om egaG6\"" }{TEXT 282 17 " is asymptotic to" }}{PARA 264 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "e^(-c)" ")%\"eG,$%\"cG!\"\"" }{TEXT -1 1 "," }}{PARA 266 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "c=card(Omega) " "/%\"cG-%%cardG6#%&OmegaG" }{TEXT -1 68 ". More generally, the proba bility that a random permutation of size " }{XPPEDIT 18 0 "n" "I\"nG6 \"" }{TEXT -1 5 " has " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 21 " \+ exceptions of type " }{XPPEDIT 18 0 "Omega" "I&OmegaG6\"" }{TEXT -1 7 " (with " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 47 " fixed) is given by a Poisson law of parameter " }{XPPEDIT 18 0 "c" "I\"cG6\"" }{TEXT -1 1 "," }}{PARA 265 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Pr(`# exce ptions`=k)=exp(-c)*c^k/k!*(1+o(1)" "/-%#PrG6#/%-#~exceptionsG%\"kG**-% $expG6#,$%\"cG!\"\"\"\"\")F.F(F0-%*factorialG6#F(F/,&\"\"\"F0-%\"oG6# \"\"\"F0F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 56 "A weaker fo rm of this property (for position gaps where " }{XPPEDIT 18 0 "Omega" "I&OmegaG6\"" }{TEXT -1 111 " is an initial segment of the integers) i s established by probabilistic arguments in [Barbour, Holst, Svanson, \+ " }{TEXT 310 21 "Poisson Approximation" }{TEXT -1 17 ", Oxford, 1992]. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 193 "The approach via hypergeometrics points to a unified derivation via gener ating functions and divergent series that encompasses more general con straints (like succession gaps and the case where " }{XPPEDIT 18 0 "Om ega" "I&OmegaG6\"" }{TEXT -1 100 " is not an initial segment of the in tegers). The principle of the generating function method is that" }} {PARA 276 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "coeff[z^n]*hypergeom( [1,1],[],z+d*z^2+O(z^3))=n!*e^d*(1+o(1))" "/*&&%&coeffG6#)%\"zG%\"nG\" \"\"-%*hypergeomG6%7$\"\"\"\"\"\"7\",(F(F**&%\"dGF**$F(\"\"#F*F*-%\"OG 6#*$F(\"\"$F*F**(-%*factorialG6#F)F*)%\"eGF4F*,&\"\"\"F*-%\"oG6#\"\"\" F*F*" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 146 "provided the arg ument of the hypergeometric is a function that is analytic at the orig in. For instance, in the case of the menage problem, one has" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "P01=convert_hypergeom(P01); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$**-%#EiG6$\"\"\",$*&,(F)F)%\"zG \"\"#*$F-F.F)F)F-!\"\"F0F),&F-F)F)F)F)F-F0-%$expG6#,$*&F1F.F-F0F0F)F0* &-%*hypergeomG6%7$F)F)7\"*&F,F0F-F)F)F1F0" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 28 "series(1/(1+2*z+z^2),z=0,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"zG\"\"\"\"\"!!\"#\"\"\"\"\"$\"\"#!\"%\"\"$\"\"&\" \"%!\"'\"\"&-%\"OG6#F%\"\"'" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 60 "so that the asymptotic proportion of menage permutations is " } {TEXT 311 8 "provedly" }{TEXT -1 10 " equal to " }{XPPEDIT 18 0 "e^(-2 )" ")%\"eG,$\"\"#!\"\"" }{TEXT -1 71 ". The method works for any of th e examples discussed in this worksheet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 271 "A companion worksheet shows that a \+ similar approach applies to \"succession gaps\", by which we means con straints on the values of successive elements in a permutation. The co rresponding solutions serve in the analysis of multiconnectivity in ra ndom interconnection graphs." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}} {MARK "33 21 0" 203 }{VIEWOPTS 1 1 0 1 1 1803 } F-!\"\"F0F),&F-F)F)F)F)F-F0-%$expG6#,$*&F1F.F-F0F0F)F0* &-%*hypergeomG6%7$F)F)7\"*&F,F0F-F)F)F1F0" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 28 "series(1/(1+2*z+z^2),z=0,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"zG\"\"\"\"\"!!\"#\"\"\"\"\"$\"\"#!\"%\"\"$\"\"&\" \"%!\"'\"\"&-%\"OG6# AutoComb,autocomb,permutationsMAD,CommonLib,FileNew#AutoComb,autocomb,hierarchy_treesq#MAD,DocumentGenerator,MathKeyword$AutoComb,autocomb,Stirling_numbersMAD,DocumentGenerator,Nth/Doc-Mgfun,Mgfun,rational_creative_telescopingy Doc-Mgfun,Holonomy,dfinite_add֌MAD,DocumentGenerator,Section"MAD,DocumentGenerator,InlineMathו+MAD,DocumentGenerator,crossref-translator 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-1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 259 30 "A SEATING ARRANGEMENT P ROBLEM\n" }}{PARA 257 "" 0 "" {TEXT 260 17 "Philippe Flajolet" }} {PARA 258 "" 0 "" {TEXT -1 29 "(Version of January 15, 1997)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "There are " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 189 " seats in a row at a l uncheonette, and people sit down one at a time at random. They are unf riendly and so never sit next to one another. What is the expected num ber of persons to sit down?" }}{PARA 0 "" 0 "" {TEXT -1 106 "The origi nal problem is due to Freedmann and Shepp, and it appeared as Problem \+ 62-3 in the 1962 volume of " }{TEXT 256 11 "SIAM Review" }{TEXT -1 247 ". There are various alternative formulations. One of them involve s fatmen that need more than one stool to sit on. Another one is a sim plified description of channel occupation for mobile telephones due to the Math. Center at Bell Labs: there are " }{XPPEDIT 18 0 "n" "I\"nG6 \"" }{TEXT -1 311 " consecutive radio channels and stations arrive at \+ random and try to grab a free channel; because of possible interferenc es, no station occupies a channel next to an already occupied one. Wha t is the expected proportion of occupied channels?\nClearly, the numbe r of occupied seats/channels lie somewhere between " }{XPPEDIT 18 0 "n /3" "*&%\"nG\"\"\"\"\"$!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "n/2 " "*&%\"nG\"\"\"\"\"#!\"\"" }{TEXT -1 99 ". This worksheet explores th e way the solution to this and similar problems may be found using the " }{HYPERLNK 17 "Gfun" 2 "gfun" "" }{TEXT -1 116 " package. The commo n schema explored here is: (1) write down an immediate specification o f the problem; (2) use the " }{HYPERLNK 17 "gfun[listtorec]" 2 "gfun[l isttorec]" "" }{TEXT -1 144 " procedure to guess the right differentia l equation; (3) exploit the results using Maple capabilities for integ ration and asymptotic expansion. " }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Basic equations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 18 0 "g(n)" "-%\"gG6#%\"nG" }{TEXT -1 8 " be the " }{TEXT 258 31 "probability generating function" }{TEXT -1 53 " (PGF) of the n umber of occupied seats when they are " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 52 " seats. In the Maple code below, we take implicitly " } {XPPEDIT 18 0 "u" "I\"uG6\"" }{TEXT -1 77 " as the generating variable . If the first individual to arrive occupies seat " }{XPPEDIT 18 0 "K " "I\"KG6\"" }{TEXT -1 78 ", then the number of occupied seats is 1 pl us the number of occupied seats in " }{XPPEDIT 18 0 "[1..K-2]" "7#;\" \"\",&%\"KG\"\"\"\"\"#!\"\"" }{TEXT -1 38 " plus the number of occupie d seats in " }{XPPEDIT 18 0 "[K+2..n]" "7#;,&%\"KG\"\"\"\"\"#F&%\"nG" }{TEXT -1 11 ", as seats " }{XPPEDIT 18 0 "K-1" ",&%\"KG\"\"\"\"\"\"! \"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "K+1" ",&%\"KG\"\"\"\"\"\"F$ " }{TEXT -1 51 " have become unavailable. The subproblems of sizes " } {XPPEDIT 18 0 "K-2" ",&%\"KG\"\"\"\"\"#!\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "n-K-1" ",(%\"nG\"\"\"%\"KG!\"\"\"\"\"F&" }{TEXT -1 56 " are of a similar nature. By the randomness assumption, " }{XPPEDIT 18 0 "K" "I\"KG6\"" }{TEXT -1 21 " takes each value in " }{XPPEDIT 18 0 "[1..n]" "7#;\"\"\"%\"nG" }{TEXT -1 32 " with equal probability, nam ely " }{XPPEDIT 18 0 "1/n" "*&\"\"\"\"\"\"%\"nG!\"\"" }{TEXT -1 55 ". \+ This gives rise to a recurrence on random variables " }{XPPEDIT 18 0 "L[n]" "&%\"LG6#%\"nG" }{TEXT -1 59 " (the number of occupied seats) a nd on generating functions" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "L[n]= 1+L[K-2]+L[n-K-1], Pr(K=k)=1/n;g(n)=u/n*Sum(g(k-2)*g(n-k-1),k=1..n),g( -1)=1,g(0)=1,g(1)=u;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"LG6#%\"nG ,(\"\"\"F)&F%6#,&%\"KGF)!\"#F)F)&F%6#,(F'F)F-!\"\"F2F)F)/-%#PrG6#/F-% \"kG*$F'F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/-%\"gG6#%\"nG*(%\"uG\" \"\"F'!\"\"-%$SumG6$*&-F%6#,&%\"kGF*!\"#F*F*-F%6#,(F'F*F3F+F+F*F*/F3;F *F'F*/-F%6#F+F*/-F%6#\"\"!F*/-F%6#F*F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "This recurrence determines the " }{XPPEDIT 18 0 "g(n)" "- %\"gG6#%\"nG" }{TEXT -1 60 " explicitly and is implemented by the fol lowing Maple code:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "g:=proc(n) l ocal k; option remember;\n if n<=0 then 1 elif n=1 then u else\n \+ expand(u/n*convert([seq(g(k-2)*g(n-k-1),k=1..n)],`+`));\n fi\n end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "seq([j,g(j)],j=0..5 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(7$\"\"!\"\"\"7$F%%\"uG7$\"\"#F'7 $\"\"$,&*$F'F)#F)F+F'#F%F+7$\"\"%F-7$\"\"&,&*$F'F+#\"\"(\"#:F-#\"\")F8 " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "For instance, when " } {XPPEDIT 18 0 "n=3" "/%\"nG\"\"$" }{TEXT -1 23 ", there is probability " }{XPPEDIT 18 0 "1/3" "*&\"\"\"\"\"\"\"\"$!\"\"" }{TEXT -1 107 " tha t just one seat is occupied: this occurs only if the first person that arrives chooses the middle seat." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "We then get the moments by successive differentiation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "subs(u=1,diff([seq(g(j),j=0..20)],u ));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#77\"\"!\"\"\"F%#\"\"&\"\"$\"\"# #\"#P\"#:#\"#E\"\"*#\"$\\$\"$0\"#\"$p\"\"#X#\"&t=\"\"%NG#\"%xs\"%v:#\" 'nv:\"&&=J#\"'\\KB\"&DD%#\")^X*>\"\"(Dq-##\")`9z?\"(DWF$#\"*fhi='\")Dh @\"*#\"*Eb42$\")DvcU#\"+J]lXv\"*DEz')*#\",)*>zEk%\"+vehYd#\".*[d+Pb<\" -D'e'Ri?#\".-)yMjoV\"-v$\\BY)[" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "evalf(subs(u=1,diff([seq(g(j)/j,j=1..40)],u)),5);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7J$\"\"\"\"\"!$\"&++&!\"&$\"&cb&F)F'$\"&L$\\F)$ \"&[\"[F)$\"&$[ZF)$\"&Wp%F)$\"&Ll%F)$\"&.i%F)$\"&Lf%F)$\"&3d%F)$\"&=b% F)$\"&b`%F)$\"&8_%F)$\"&*3XF)$\"&!)\\%F)$\"&$)[%F)$\"&'zWF)$\"&=Z%F)$ \"&[Y%F)$\"&$eWF)$\"&DX%F)$\"&rW%F)$\"&@W%F)$\"&wV%F)$\"&LV%F)$\"&%HWF )$\"&dU%F)$\"&BU%F)$\"&\">WF)$\"&hT%F)$\"<%F)$\"&2T%F)$\"WF)$\"&e S%F)$\"&OS%F)$\"&:S%F)$\"&&*R%F)$\"&wR%F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "This suggests that the mean occupation ration could be, f or " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 60 " large, asymptotic to a constant with approximate value 44%." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "The mean occupancy ratio" }}{PARA 0 "" 0 "" {TEXT -1 226 "The easiest is to try a heuristic approach. As the recurrences for mo ments are linear, it is reasonable to expect them to be of the holonom ic type. We thus compute a few dozen initial values and try to guess a recurrence with " }{HYPERLNK 17 "gfun[listtorec]" 2 "gfun[listtorec] " "" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "with( gfun);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7T%(LaplaceG%.algebraicsubsG %.algeqtodiffeqG%.algeqtoseriesG%.algfuntoalgeqG%&borelG%.cauchyproduc tG%.diffeq*diffeqG%.diffeq+diffeqG%,diffeqtorecG%)guesseqnG%(guessgfG% 0hadamardproductG%0holexprtodiffeqG%)invborelG%,listtoalgeqG%-listtodi ffeqG%0listtohypergeomG%+listtolistG%.listtoratpolyG%*listtorecG%-list toseriesG%5listtoseries/LaplaceG%1listtoseries/egfG%4listtoseries/lgde gfG%4listtoseries/lgdogfG%1listtoseries/ogfG%4listtoseries/revegfG%4li sttoseries/revogfG%,maxdegcoeffG%*maxdegeqnG%,maxordereqnG%,mindegcoef fG%*mindegeqnG%,minordereqnG%*optionsgfG%,poltodiffeqG%)poltorecG%/rat polytocoeffG%(rec*recG%(rec+recG%,rectodiffeqG%*rectoprocG%.seriestoal geqG%/seriestodiffeqG%2seriestohypergeomG%-seriestolistG%0seriestoratp olyG%,seriestorecG%/seriestoseriesG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "rec:=listtorec(subs(u=1,diff([seq(g(j),j=0..25)],u)), u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$recG7$<&,*-%\"uG6#%\"nG! \"#*&,&F+!\"\"\"\"\"F0F0-F)6#,&F+F0F0F0F0F0*&,&F+\"\"#\"\"%F0F0-F)6#,& F+F0F6F0F0F0*&,&F+F/!\"$F0F0-F)6#,&F+F0\"\"$F0F0F0/-F)6#\"\"!FE/-F)6#F 0F0/-F)6#F6F0%$ogfG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "The recurr ence transforms into a differential equation by means of " }{HYPERLNK 17 "gfun[rectodiffeq]" 2 "gfun[rectodiffeq]" "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ode:=rectodiffeq(op(1,rec),u(n),Y(z));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$odeG<$,(*&,&*$%\"zG\"\"#F+F*!\"#\"\"\"-%\"YG6 #F*F-F-*&,(F)F-F*F,F-F-F--%%diffG6$F.F*F-F-!\"\"F-/-F/6#\"\"!F:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 149 "Now, we are sure of the existence of a closed form for the generating function of averages, since any O DE of order 1 is solvable by quadratures. The " }{HYPERLNK 17 "dsolve " 2 "dsolve" "" }{TEXT -1 31 " command of Maple does the job:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "M1_z:=factor(op(2,dsolve(ode,Y(z))) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%M1_zG,$*(,&-%$expG6#,$%\"zG\" \"#\"\"\"!\"\"F.F.-F)6#,$F,!\"#F.,&F/F.F,F.F3#F.F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "We can check consistency with known values " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "series(M1_z,z=0,30);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#+in%\"zG\"\"\"\"\"\"F%\"\"##\"\"&\"\"$\"\"$\"\"# \"\"%#\"#P\"#:\"\"&#\"#E\"\"*\"\"'#\"$\\$\"$0\"\"\"(#\"$p\"\"#X\"\")# \"&t=\"\"%NG\"\"*#\"%xs\"%v:\"#5#\"'nv:\"&&=J\"#6#\"'\\KB\"&DD%\"#7#\" )^X*>\"\"(Dq-#\"#8#\")`9z?\"(DWF$\"#9#\"*fhi='\")Dh@\"*\"#:#\"*Eb42$\" )DvcU\"#;#\"+J]lXv\"*DEz')*\"#<#\",)*>zEk%\"+vehYd\"#=#\".*[d+Pb<\"-D' e'Ri?\"#>#\".-)yMjoV\"-v$\\BY)[\"#?#\"0z8Ni*oaO\"/D,[&Hz*Q\"#@#\"1[ZHM oJ,5\"0v$>5j)3-\"\"#A#\"2JQHTJ*[!f%\"1vV@!)*=E[%\"#B#\"2/Bv0T7W'o\"1D1 AaPeJk\"#C#\"5VjC\\LHN(*o8\"4DJ&*e&>-sK7\"#D#\"5%p4(Hw+\"Q(=A\"4vo'>^U N/B>\"#E#\"5\\*=v*e58iIq\"4DJ&zU\"G\\N(e\"#F#\"6;p\"*4)[`]M%z$\"5v$4J' *RX(QfI\"#G#\"9\"H7BY)QQHfTJ[\"8D\"y:vrYMxNkP\"#H-%\"OG6#F%\"#I" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "We can be also quite sure that th e process makes sense if we compare as well with values that haven't \+ been used at all in the \"guessing\" phase:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "subs(u=1,diff([seq(g(j),j=26..29)],u));" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&#\"5%p4(Hw+\" Q(=A\"4vo'>^UN/B>#\"5\\*=v*e58iIq\"4DJ&zU\"G\\N(e#\"6;p\"*4)[`]M%z$\"5 v$4J'*RX(QfI#\"9\"H7BY)QQHfTJ[\"8D\"y:vrYMxNkP" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "The generating function of expectations is meromorph ic with only a pole at " }{XPPEDIT 18 0 "z=1" "/%\"zG\"\"\"" }{TEXT -1 124 ". In order to analyse the coefficients of the explicit solutio n found, we examine the singular expansion at the double pole " } {XPPEDIT 18 0 "z=1" "/%\"zG\"\"\"" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 31 "map(normal,series(M1_z,z=1,4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+-,&!\"\"\"\"\"%\"zGF&,$*&,&-%$expG6#\"\"#F&F%F &F&-F,6#!\"#F&#F&F.!\"#F/!\"\",$F/F%\"\"!,$F/#F.\"\"$\"\"\"-%\"OG6#F& \"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 168 "By the principles of si ngularity analysis, it is enough to expand the singular part. This sho ws that the mean number of occupied seats satisfies the approximate fo rmula" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "m1:=1/2*(1-exp(-2) )*(n+1)-exp(-2); evalf(m1,20); C1:=evalf(coeff(m1,n,1)):" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#m1G,&*&,&\"\"\"F(-%$expG6#!\"#!\"\"F(,&%\"nGF (F(F(F(#F(\"\"#F)F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"nG$\"51aOp \"QeBLK%!#?$\"5 " 0 "" {MPLTEXT 1 0 78 "for j from 0 to 30 by 5 do j,evalf(subs(u=1,diff(g(j),u))-subs(n=j ,m1),30) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!$!?Uvv+!f@'43X^2( *pH!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"&$\"=blkkECMu68'*z2!)!#I " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#5$!:lB#eoG_=dXk)>$!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#:$\"5)o,*3rNG-!\\\"!#H" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$\"#?$!0!*Rs'G#zd\"!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#D$\"*UAiO&!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#I$!$E(!#H " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "In particular the constant fo und empirically to be close to 0.44 is precisely" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "1/2*(1-exp(-2))=evalf(1/2*(1-exp(-2)),30);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&#\"\"\"\"\"#F&-%$expG6#!\"##!\"\"F '$\"?9DD+I0aOp\"QeBLK%!#I" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "Di stributional analysis" }}{PARA 0 "" 0 "" {TEXT -1 217 "Whenever possib le in analysis of algorithms, one should try to determine how characte ristic the average case is. We show now that the standard deviation of the distribution of the number of occupied seats/channels is " } {XPPEDIT 18 0 "O(sqrt(n))" "-%\"OG6#-%%sqrtG6#%\"nG" }{TEXT -1 67 ". B y the Markov-Chebyshev inequalities, this means that, for large " } {XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 73 ", almost all configurations must be close to the average predicted value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "First, we have acc ess to the second (factorial) moments by a double differentiation." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "l2:=subs(u=1,diff([seq(g(j) ,j=0..25)],u,u));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#l2G7<\"\"!F&F& #\"\"%\"\"$\"\"##\"#e\"#:#\"#]\"\"*#\"$y#\"#N#\"$e\"F-#\"&%\\Q\"%NG#\" &;n#\"%v:#\"')eY'\"&&=J#\"'e:@\"%0&)#\"'Qc%*\"&v@$#\")#\\:u$\"(v94\"# \"+C$epg$\")Dh@\"*#\"*'3VZF\"(v53'#\"-e!)ficb\",v)=Z&3\"#\",%HCH5j)3-\"#\"3AT#)\\]IDjU\"1vV@!)*=E[%#\"2M*pCDC@lg\" 0v=?K%*o%e#\"4A08(>CQn " 0 "" {MPLTEXT 1 0 65 "gfun['maxordereqn'],gfun['maxdegcoeff'];\nrec:=listto rec(l2,u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$recG%%FAILG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "The control parameters can be set to higher values. " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "gfun['maxordereqn']:=8; gfun['maxde gcoeff']:=6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%gfunG6#%,maxordere qnG\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%gfunG6#%,maxdegcoeffG \"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "We can now determine the right recurrence (we need more values):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "l2:=subs(u=1,diff([seq(g(j),j=0..55)],u,u)):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "rec2:=listtorec(l2,u(n));" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%rec2G7$<*,2*&,&\"#[\"\"\"%\"nG\"#C F+-%\"uG6#F,F+F+*&,(!#KF+F,\"#A*$F,\"\"#\"#=F+-F/6#,&F,F+F+F+F+F+*&,* \"#GF+F,!#hF5!#=*$F,\"\"$FAF+-F/6#,&F,F+F6F+F+F+*&,*!#7F+F,\"#8F5F?F@! \"(F+-F/6#,&F,F+FAF+F+F+*&,*!$_#F+F,!#()F5\"#9F@\"\"&F+-F/6#,&F,F+\"\" %F+F+F+*&,*\"#?F+F,!#TF5!#CF@!\"$F+-F/6#,&F,F+FRF+F+F+*&,*\"$![F+F,\"$ 'HF5\"#gF@FVF+-F/6#,&F,F+\"\"'F+F+F+*&,*!$!GF+F,!$m\"F5F3F@!\"#F+-F/6# ,&F,F+\"\"(F+F+F+/-F/6#FR#\"#e\"#:/-F/6#Fbo#\"#]\"\"*/-F/6#\"\"!F[q/-F /6#F+F[q/-F/6#F6F[q/-F/6#FA#FVFA/-F/6#FVF6%$ogfG" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 218 "The recurrence is of order 7. The generating func tion satisfies a differential equation of order 3 with coefficients of degree 7 (!!). It is rather remarkable that the dsolve command of Map le can solve this explicitly." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "od e2:=rectodiffeq(op(1,rec2),u(n),Y(z));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%ode2G<*/-%\"YG6#\"\"!F*/--%\"DG6#F(F)F*/---%#@@G6$F.\"\"$F/F) \"\")/---F46$F.\"\"'F/F)\"%+S/---F46$F.\"\"%F/F)\"#[/---F46$F.\"\"#F/F )F*/---F46$F.\"\"&F/F)\"$k%,**&,**$%\"zGF=!%S9*$FWFQ\"%!3\"*$FWFD!%?;* $FWF6\"$?(\"\"\"-F(6#FWFinFin*&,.FY!$?%FV!$?\"*$FW\"\"(!$?(*$FWFK\"$S# Fen\"$+*\"$?\"FinFin-%%diffG6$FjnFWFinFin*&,.F`o!$S&FV\"$5)FYFboFen\"$ I*Fgn!$g$FWF_oFin-Fho6$FgoFWFinFin*&,.F`o!#!*FV\"$5#FY!$]\"Fen\"#!*Fgn F_oFco\"#gFin-Fho6$F`pFWFinFin" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Note that a simple rational solution is detected by dsolve. This enta ils a reduction of order," }}{PARA 0 "" 0 "" {TEXT -1 92 "and a comple te algorithm exists for order 2 (in fact that Maple succeeds in bypass ing here)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "infolevel[dsolve]:=5: M2_z:=factor(op(2,dsolve(ode2,Y(z))));" }}{PARA 6 "" 1 "" {TEXT -1 421 "dsolve/diffeq/polylinearODE: checking Euler equation\ndsolve/di ffeq/expsols: trying exponential solutions\ndsolve/diffeq/expsols: \+ rational solutions partially successful. Result(s)= (1-3*z)/(-1+z)^ 3\ndsolve/diffeq/expsols_solvericcati: all solutions by polynomial p art\ndsolve/diffeq/expsols: expon. solutions partially successful. R esult(s) = exp(Int((-2*z^2+z-2)/(z^2-z),z)), exp(Int((-4*z^2-2*z)/(- 1+z^2),z))" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%M2_zG,$*&,,\"\"\"F(% \"zG!\"$*&-%$expG6#,$F)!\"#F(F)\"\"#\"\"%-F-6#,$F)!\"%!\"\"*&F3F(F)F(F 7F(,&F7F(F)F(F*#F(F2" }}}{PARA 0 "" 0 "" {TEXT -1 22 " The singular pa rt at " }{XPPEDIT 18 0 "z=1" "/%\"zG\"\"\"" }{TEXT -1 66 " is analysed . For the variance this leads to approximate formulae." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Moment2_sing:=map(simplify,series(M 2_z+M1_z,z=1,3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-Moment2_singG+ +,&!\"\"\"\"\"%\"zGF(,(#F'\"\"#F(-%$expG6#!\"%F+-F.6#!\"#F(!\"$,$*&,(! \"(F(-F.6#\"\"%F(-F.6#F,F,F(F-F(#F'F;!\"#,$F-!\"$!\"\"-%\"OG6#F(\"\"! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "var_n_asympt:=factor(c ollect(expand(convert([seq((-1)^j*coeff(Moment2_sing,z-1,j)*binomial(n -j-1,-j-1),j=-3..-1)],`+`)-m1^2),n,simplify));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-var_n_asymptG*&-%$expG6#!\"%\"\"\",&%\"nGF*\"\"$F*F* " }}}{PARA 0 "" 0 "" {TEXT -1 45 "Again, the approximations are extrem ely good:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalf(var_n_asy mpt,30);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"nG$\"?KF@!=PH!=M()))Q cJ=!#J$\"?'>Q1a6)3a-im;p%\\&F'\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "evalf(subs(n=50,var_n_asympt),30); evalf(subs(u=1,(d iff(g(50),u,u)+diff(g(50),u)-diff(g(50),u)^2)),30);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"?![F^0nbb6H5h)G2(*!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?#[F^0nbb6H5h)G2(*!#I" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "We have in passing obtained a " }{TEXT 262 7 "Theorem" } {TEXT -1 2 ". " }{TEXT 263 91 "In the random seating problem, the vari ance of the number of occupied seats when there are " }{XPPEDIT 264 0 "n" "I\"nG6\"" }{TEXT 265 24 " seats is asymptotic to" }}{PARA 259 " " 0 "" {XPPEDIT 18 0 "(n+3)/exp(4)=.0183156388887341802937180212732*n+ .054946916666202540881154063820" "/*&,&%\"nG\"\"\"\"\"$F&F&-%$expG6#\" \"%!\"\",&*&$\"?KF@!=PH!=M()))QcJ=!#JF&F%F&F&$\">?Q1a6)3a-im;p%\\&!#IF &" }{TEXT -1 0 "" }{TEXT -1 1 "." }}}{PARA 0 "" 0 "" {TEXT -1 153 "Thi s result seems to be new (!). Convergence is extremely fast so that th is formula is highly accurate. The standard deviation is found to be o nly about " }{XPPEDIT 18 0 "sqrt(n)/7" "*&-%%sqrtG6#%\"nG\"\"\"\"\"(! \"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 261 12 "Distribution" }{TEXT -1 17 ". A mean that is " } {XPPEDIT 18 0 "O(n)" "-%\"OG6#%\"nG" }{TEXT -1 34 " and a standard dev iation that is " }{XPPEDIT 18 0 "O(sqrt(n))" "-%\"OG6#-%%sqrtG6#%\"nG " }{TEXT -1 192 " entail that the distribution is concentrated around \+ its mean with high probability. This also suggests that the distributi ons of the number of occupied seats could be asymptotically Gaussian. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "distr:=sort(map(proc(e) [op([2,2],e),op(1,e)] end,[op(evalf(g(60),4))]),proc(x,y) evalb(op(1, x) " 0 "" {MPLTEXT 1 0 33 "linalg [transpose](matrix(distr));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATR IXG6#7$7-\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G\"#H\"#I7-$\"%oG!#8$\"%T' )!#5$\"%q7!\"($\"%(4%!\"'$\"%;W!\"&$\"%2>!\"%$\"%nNFE$\"%CHFE$\"%]**FB $\"%+7FB$\"%[LF<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(di str,style=POINT);" }}{PARA 13 "" 1 "" {INLPLOT "6$-%'CURVESG6$7-7$$\"# ?\"\"!$\"3#)***********z'G!#F7$$\"#@F*$\"3:++++++T')!#C7$$\"#AF*$\"3(* ************p7!#@7$$\"#BF*$\"3&************p4%!#?7$$\"#CF*$\"3z******* ****fT%!#>7$$\"#DF*$\"33++++++2>!#=7$$\"#EF*$\"3<++++++nNFK7$$\"#FF*$ \"3$************R#HFK7$$\"#GF*$\"3^++++++]**FE7$$\"#HF*$\"3-+++++++7FE 7$$\"#IF*$\"31++++++[LF9-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-%&STYLEG6#%&P OINTG" 2 264 264 264 5 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{PARA 0 " " 0 "" {TEXT -1 138 "In 65% of the cases, the occupation is either 26 \+ or 27; the probability of an extremely bad assignment (20 seats out of 60) is only about " }{XPPEDIT 18 0 "3*10^(-9)" "*&\"\"$\"\"\")\"#5,$ \"\"*!\"\"F$" }{TEXT -1 0 "" }{TEXT -1 152 ". In fact, a Gaussian law \+ can be proved by adapting the bivariate analysis of patterns in binary search trees by Flajolet, Martinez, and Gourdon (1996)." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Fatter men" }}{PARA 0 "" 0 "" {TEXT -1 51 "The approach extends to the case where fatmen need " }{XPPEDIT 18 0 "b" "I\"bG6\"" }{TEXT -1 42 " seats on each side. The earlier case w as " }{XPPEDIT 18 0 "b=1" "/%\"bG\"\"\"" }{TEXT -1 19 ". We consider h ere " }{XPPEDIT 18 0 "b=2" "/%\"bG\"\"#" }{TEXT -1 65 ". This time, th e number of occupied seats lies somewhere between " }{XPPEDIT 18 0 "n/ 3" "*&%\"nG\"\"\"\"\"$!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "n/4" "*&%\"nG\"\"\"\"\"%!\"\"" }{TEXT -1 14 ". Now, we let " }{XPPEDIT 18 0 "gb" "I#gbG6\"" }{TEXT -1 0 "" }{TEXT -1 51 " be the probability gen erating function (PGF) with " }{XPPEDIT 18 0 "u" "I\"uG6\"" }{TEXT -1 0 "" }{TEXT -1 60 " the generating variable. The following procedure \+ computes " }{XPPEDIT 18 0 "gb" "I#gbG6\"" }{TEXT -1 18 " for a given s ize " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 19 " and the parameter \+ " }{XPPEDIT 18 0 "b" "I\"bG6\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "gb:=proc(n,b) local k; option remember;\n if n<=0 then 1 elif n<=b then u else expand(u/n*convert([seq(gb(k-b-1,b) *gb(n-k-b,b),k=1..n)],`+`))\n fi\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "The probability generating functions are now:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "seq([j,gb(j,2)],j=0..6);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6)7$\"\"!\"\"\"7$F%%\"uG7$\"\"#F'7$\"\"$F'7$\"\"%,&*$F' F)#F%F)F'F07$\"\"&,&F/#F-F2F'#F%F27$\"\"'F/" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 18 "For instance, for " }{XPPEDIT 18 0 "n=4" "/%\"nG\"\"%" }{TEXT -1 83 ", we have 2 occupied seats if the first arrival is on a \+ side (this has probability " }{XPPEDIT 18 0 "1/2" "*&\"\"\"\"\"\"\"\"# !\"\"" }{TEXT -1 75 "), which leaves the opposite seat available, and \+ 1 occupied seat otherwise." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "The moments are obtained by differentiation of the PGF:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "subs(u=1,diff([seq(gb(j,2),j=0..25)],u)); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7<\"\"!\"\"\"F%F%#\"\"$\"\"##\"\"* \"\"&F(#\"#;\"\"(#\"$.\"\"#S#\"$G\"\"#X#\"%\"4\"\"$]$#\"#&*\"#G#\"&`a& \"&?^\"#\"'TGK\"&+>)#\"'\"*=P\"&+#))#\"'tPO\"&+5)#\"(nP\\(\"(![s:#\"*V \"HZH\")+mZe#\"+Z4)f$\\\"*+guG*#\",^7&[$Q%\"++SuUy#\",*>NMIj\",++o&z5# \"-.A+QcK\",?:(*\\I&#\"/LIi$\\%=C\".+?ZR7x$#\"1(oz$)[Pc%=\"0++Ohh)fF# \"2F>iTc$R5U\"1++G`npZg#\"2R%yUGNVdQ\"1+++Y>]I`" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 52 "evalf(subs(u=1,diff([seq(gb(j,2)/j,j=1..35)],u )),5);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7E$\"\"\"\"\"!$\"&++&!\"&$\" &LL$F)$\"&+v$F)$\"&+g$F)F*$\"&`E$F)$\"&)=KF)$\"&0;$F)$\"&r6$F)$\"&W3$F )$\"&j0$F)$\"&A.$F)$\"&=,$F)$\"&S*HF)$\"&&yHF)$\"&['HF)$\"&E&HF)$\"&<% HF)$\"&>$HF)$\"&I#HF)$\"&\\\"HF)$\"&w!HF)$\"&3!HF)$\"&Y*GF)$\"&*))GF)$ \"&O)GF)$\"&'yGF)$\"&S(GF)$\"&)pGF)$\"&e'GF)$\"&?'GF)$\"&&eGF)$\"&^&GF )$\"&?&GF)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "This suggests an oc cupation ratio of about 28%, now." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Like before, we can guess a differential equation and attempt to s olve it." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "lb1:=subs(u=1, diff([seq(gb(j,2),j=0..35)],u)): gfun['maxordereqn']:=5; gfun['maxdegc oeff']:=5; recb:=listtorec(lb1,u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%gfunG6#%,maxordereqnG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> &%%gfunG6#%,maxdegcoeffG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%re cbG7$<'/-%\"uG6#\"\"!F+/-F)6#\"\"\"F//-F)6#\"\"#F//-F)6#\"\"$F/,,-F)6# %\"nGF3-F)6#,&F;F/F/F/!\"#*&,&F;F/F3F/F/-F)6#FAF/F/*&,&!\"'F/F;F?F/-F) 6#,&F;F/F7F/F/F/*&,&F;F/\"\"%F/F/-F)6#FKF/F/%$ogfG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "rectodiffeq(op(1,recb),u(n),Y(z));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<$/-%\"YG6#\"\"!F(,(*&,&*$%\"zG\"\"$! \"%*$F-\"\"#\"\"%\"\"\"-F&6#F-F3F3*&,(F0!\"#F-F2F8F3F3-%%diffG6$F4F-F3 F3F1F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "The differential equati on is of first order, hence again solvable by quadratures" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 21 "solb:=dsolve(\",Y(z));" }}{PARA 6 "" 1 "" {TEXT -1 216 "dsolve/diffeq/dsol1: -> first order, first degree meth ods :\ndsolve/diffeq/dsol1: trying linear bernoulli\ndsolve/diffeq/l inearsol: solving 1st order linear d.e.\ndsolve/diffeq/dsol1: line ar bernoulli successful" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solbG/-% \"YG6#%\"zG,$*(,&**%\"IG\"\"\"-%$erfG6#,&*&F.F/F)F/F/F.F/F/%#PiGF/-%$e xpG6#,$*$,&F/F/F)F/\"\"#!\"\"F/F/*,F.F/-F76#,$*&F)F/,&F)F/F " 0 "" {MPLTEXT 1 0 32 "singb:=series(op(2,solb),z=1,3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&singbG++,&!\"\"\"\"\"%\"zGF(,$*&,&**%\"IGF(-%$erfG6# ,$F.\"\"#F(%#PiGF(-%$expG6#!\"%F(F(*,F.F(-F66#!\"$F(F4F(-F06#F.F(-F66# F'F(F'F(F4#F'F3FA!\"#,$*&,(F-F8*(F4#F(F3-F66#\"\"%F(F5F(!\"#F9FJF(F4FA FA!\"\",$*&,(F-\"\"(FFFJF9!\"(F(F4FAFA\"\"!-%\"OG6#F(\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "c2:=factor(simplify(coeff(s ingb,z-1,-2))); c1:=factor(simplify(coeff(singb,z-1,-1))); C2:=evalf(c 2):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2G,$**%\"IG\"\"\"%#PiG#F(\" \"#-%$expG6#!\"%F(,&-%$erfG6#,$F'F+!\"\"-F26#F'F(F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c1G,$*(%\"IG\"\"\",(*(-%$erfG6#,$F'\"\"#F(-%$ex pG6#!\"%F(%#PiGF(!\"#*&F'F(F4#F(F/F(*(-F,6#F'F(F0F(F4F(F/F(F4#!\"\"F/F <" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "This corresponds to an asymp totic form for the first moment" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " mb1:=c2*(n+1)-c1; evalf(mb1,30);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% $mb1G,&*,%\"IG\"\"\"%#PiG#F(\"\"#-%$expG6#!\"%F(,&-%$erfG6#,$F'F+!\"\" -F26#F'F(F(,&%\"nGF(F(F(F(F**(F'F(,(*(F1F(F,F(F)F(!\"#*&F'F(F)F*F(*(F6 F(F,F(F)F(F+F(F)#F5F+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"nG$\"? -EHdNjX%zds()4bu#!#I$\"?5IY'yn\"Gs*)G'Q\\vs$F'\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "Once more, the asymptotic approximation is extr emely good, even for relatively small values of n." }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 82 "for j from 0 to 30 by 5 do j,evalf(subs(u=1,diff(gb (j,2),u))-subs(n=j,mb1),30) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\" \"!$!?5IY'yn\"Gs*)G'Q\\vs$!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"& $\"?A)R2FWmVb?UF7!\\a!#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#5$!>y)= `;ikTX$Que>7\"!#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#:$\";AKR)yooyd VUo$\\!#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#?$\"8A>-$HO3%o%4E=!#J " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#D$!6y%Rg$foFRjI$!#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#I$\"3AW?$f!))*R9%!#J" }}}{PARA 0 "" 0 "" {TEXT -1 170 "This last example demonstrates the interest of preservin g initial conditions whenever possible. The way Gfun and Maple manage \+ them consistently is especially useful here." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Fatter men, even!" }}{PARA 0 "" 0 "" {TEXT -1 188 "We \+ follow the same schema and consider finally the situation where 3 seat s/channels are unavailable next to an occupied seat. The number of occ upied seats must now lie between n/4 and n/5." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "evalf(subs(u=1,diff([seq(gb(j,3)/j,j=1..35)],u)) ,5);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7E$\"\"\"\"\"!$\"&++&!\"&$\"&L L$F)$\"&+]#F)$\"&+!GF)$\"&yx#F)$\"&Jl#F)F,$\"&WW#F)$\"<#F)$\"&TQ#F)$ \"&0N#F)$\"&BK#F)$\"&(*H#F)$\"&4G#F)$\"&TE#F)$\"&\"\\AF)$\"&dB#F)$\"&Q A#F)$\"&J@#F)$\"&N?#F)$\"&Y>#F)$\"&m=#F)$\"&#z@F)$\"&D<#F)$\"&i;#F)$\" &/;#F)$\"&]:#F)$\"&+:#F)$\"&`9#F)$\"&59#F)$\"&p8#F)$\"&I8#F)$\"&%H@F)$ \"&g7#F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "This suggests an occu pation ratio of about 21%." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "lb1:=subs(u=1,diff([seq(gb(j,3),j=0..35)],u)): gfun['maxordereqn' ]:=6; gfun['maxdegcoeff']:=3; recb:=listtorec(lb1,u(n));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%%gfunG6#%,maxordereqnG\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%gfunG6#%,maxdegcoeffG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%recbG7$<(,,-%\"uG6#%\"nG\"\"#-F)6#,&F+\"\"\"F0F0!\"# *&,&F+F0\"\"$F0F0-F)6#F3F0F0*&,&!\")F0F+F1F0-F)6#,&F+F0\"\"%F0F0F0*&,& F+F0\"\"&F0F0-F)6#F?F0F0/-F)6#\"\"!FF/-F)6#F0F0/-F)6#F,F0/-F)6#F4F0/-F )6#F=F0%$ogfG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "gfun['maxd egcoeff']:=1; rectodiffeq(op(1,recb),u(n),Y(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%gfunG6#%,maxdegcoeffG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$,(*&,&*$%\"zG\"\"%!#7*$F(\"\"$\"#7\"\"\"-%\"YG6#F(F.F .*&,(*$F(\"\"#!\"'F(F-F6F.F.-%%diffG6$F/F(F.F.\"\"'F./-F06#\"\"!F>" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "The solution now involves integra ls of cubic polynomials." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "solb:=dsolve(\",Y(z)); singb:=series(op(2,solb),z=1,3);" }}{PARA 6 " " 1 "" {TEXT -1 216 "dsolve/diffeq/dsol1: -> first order, first degr ee methods :\ndsolve/diffeq/dsol1: trying linear bernoulli\ndsolve/d iffeq/linearsol: solving 1st order linear d.e.\ndsolve/diffeq/dsol1: linear bernoulli successful" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%s olbG/-%\"YG6#%\"zG*(-%$intG6$-%$expG6#,$*&%\"uG\"\"\",(*$F3\"\"#F7F3\" \"$\"\"'F4F4#F4F8/F3;\"\"!F)F4-F/6#,$*&F)F4,(*$F)F7F7F)F8F9F4F4#!\"\"F 8F4,(FCF4F)!\"#F4F4FE" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&singbG++,& !\"\"\"\"\"%\"zGF(*&-%$intG6$-%$expG6#,$*&%\"uGF(,(*$F3\"\"#F6F3\"\"$ \"\"'F(F(#F(F7/F3;\"\"!F(F(-F/6##!#6F7F(!\"#,&F*!\"'*&-F/6##\"#6F7F(F= F(F(!\"\",&F*\"#:FD!\"$\"\"!-%\"OG6#F(\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "c2:=factor(simplify(coeff(singb,z-1,-2))); c1:= factor(simplify(coeff(singb,z-1,-1))); C3:=evalf(c2):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2G*&-%$intG6$-%$expG6#,$*&%\"uG\"\"\",(*$F.\" \"#F2F.\"\"$\"\"'F/F/#F/F3/F.;\"\"!F/F/-F*6##!#6F3F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c1G,&*&-%$intG6$-%$expG6#,$*&%\"uG\"\"\",(*$F/\" \"#F3F/\"\"$\"\"'F0F0#F0F4/F/;\"\"!F0F0-F+6##!#6F4F0!\"'F0F0" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "mb1:=c2*(n+1)-c1; evalf(mb1, 30);" }}{PARA 0 "" 0 "" {TEXT -1 119 "In particular, the mean occupati o ratio is an interesting integral that evaluates to .2009733699788442 43166574354875..." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$mb1G,(*(-%$int G6$-%$expG6#,$*&%\"uG\"\"\",(*$F/\"\"#F3F/\"\"$\"\"'F0F0#F0F4/F/;\"\"! F0F0-F+6##!#6F4F0,&%\"nGF0F0F0F0F0*&F'F0F:F0F5!\"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"nG$\"?v[Nul;VU%)y*pL(4?!#I$\">8%[?g;-(4>&)*e8 oS!#H\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "And finally, the a pproximation is still very good, being within 1% already for " } {XPPEDIT 18 0 "n=5" "/%\"nG\"\"&" }{TEXT -1 0 "" }{TEXT -1 67 ", altho ugh the asymptotic regime takes a little longer to establish" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "for j from 0 to 30 by 5 do j,evalf(subs(u =1,diff(gb(j,3),u))-subs(n=j,mb1),30) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!$!>8%[?g;-(4>&)*e8oS!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"&$!=]eA*))*z\"48Y(R/o6!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#5$!s,M!3mA!#H" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 257 11 "Conclusion s" }}{PARA 0 "" 0 "" {TEXT -1 338 "Our purpose here has been to demons trate how one naturally arrives at the solution of a probabilistic pro blem using tools like Gfun. Once the solutions have been \"guessed\", \+ it is possible to come back, think, and prove solutions. For instance, the problem of the mean leads to recurrences that involve history (su mmation) and a factor of " }{XPPEDIT 18 0 "1/n" "*&\"\"\"\"\"\"%\"nG! \"\"" }{TEXT -1 532 "; thus, it is reasonable to expect to be within r each of the theory of holonomic functions on which Gfun is based, and \+ rough bounds on the order of recurrences or differential equatiosn suf fice to validate the \"guesses\". In this way, we have \"naturally\" \+ rediscovered a solution of the generalized problem due to David Rothma n (fatter men) and obtained a variance analysis for the basic problem \+ that appears to be new. The whole session (including the variance comp utations) takes about 60 seconds of CPU time on a DEC-3000 station." } }{PARA 0 "" 0 "" {TEXT -1 110 "About the original problem, we may comp are the mean seat occuptation to the best possible seating arrangement :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "for i from 1 to 3 do `C `.i/(1/(i+1)) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+orkY')!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+J'HlB)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'zM*Q!)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 183 "Thus, \+ we have determined the price to be paid for random access: it is abou t 15% to 20%. As we saw repeatedly, the asymptotic approximations obta ined are extremely good, already for " }{XPPEDIT 18 0 "n=5" "/%\"nG\" \"&" }{TEXT -1 162 ". Also the distributional analysis, where a small \+ variance is obtained, shows that the average-case is highly representa tive of what will be observed in practice." }}}}{MARK "0 2 0" 0 } {VIEWOPTS 1 1 0 3 2 1804 } m \+ that appears to be new. The whole session (including the variance comp utations) takes about 60 seconds of CPU time on a y 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0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 49 "An Integral of a Product \+ of four Bessel Functions" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 19 "" 0 "" {TEXT -1 15 "Fr\351d\351ric Chyzak" }}{PARA 257 "" 0 "" {TEXT -1 28 "(Version of January 8, 1998)" }}{PARA 258 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "In (Glasser, M. L. and Montal di E. (1994): Some Integrals Involving Bessel Functions, " }{TEXT 259 20 "J. Math. Anal. Appl." }{TEXT -1 2 ", " }{TEXT 276 3 "183" }{TEXT -1 179 ":577-590), Glasser and Montaldi compute a closed form for an i ntegral of a product of two Bessel functions, and suggest that their t reatment should extend to the following example" }}}{EXCHG {PARA 256 " " 0 "" {XPPEDIT 18 0 "int(x*J[1](a*x)*I[1](a*x)*Y[0](x)*K[0](x),x=0..i nfinity)=-ln(1-a^4)/2/Pi/a^2" "/-%$intG6$*,%\"xG\"\"\"-&%\"JG6#\"\"\"6 #*&%\"aGF(F'F(F(-&%\"IG6#\"\"\"6#*&F0F(F'F(F(-&%\"YG6#\"\"!6#F'F(-&%\" KG6#F<6#F'F(/F';F<%)infinityG,$**-%#lnG6#,&\"\"\"F(*$F0\"\"%!\"\"F(\" \"#FO%#PiGFO*$F0\"\"#FOFO" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 255 "which is of interest because it contains each of the fou r types of Bessel functions. This integral is one of numerous integra ls containing four (or more) Bessel functions. See for instance (Prud nikov, A. P., Brychkov, Yu. A. and Marichev, O. I. (1986): " }{TEXT 260 49 "Integrals and Series. Volume 2: Special functions" }{TEXT -1 35 ", Gordon and Breach; Sec. 2.16.47)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "In this session, we deal with the integral above and deri ve a closed form for it using our " }{HYPERLNK 17 "Mgfun" 2 "Mgfun" " " }{TEXT -1 1 " " }{TEXT -1 44 "package in an intimate interaction wit h the " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 1 " " }{TEXT -1 8 " package." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(Mgfun);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#7(%,diag_of_sysG%+int_of_sysG%+pol_to _sysG%+sum_of_sysG%(sys*sysG%(sys+sysG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "with(gfun);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7V%(La placeG%.algebraicsubsG%.algeqtodiffeqG%.algeqtoseriesG%.algfuntoalgeqG %&borelG%.cauchyproductG%.diffeq*diffeqG%.diffeq+diffeqG%2diffeqtohomd iffeqG%,diffeqtorecG%)guesseqnG%(guessgfG%0hadamardproductG%0holexprto diffeqG%)invborelG%,listtoalgeqG%-listtodiffeqG%0listtohypergeomG%+lis ttolistG%.listtoratpolyG%*listtorecG%-listtoseriesG%5listtoseries/Lapl aceG%1listtoseries/egfG%4listtoseries/lgdegfG%4listtoseries/lgdogfG%1l isttoseries/ogfG%4listtoseries/revegfG%4listtoseries/revogfG%,maxdegco effG%*maxdegeqnG%,maxordereqnG%,mindegcoeffG%*mindegeqnG%,minordereqnG %*optionsgfG%,poltodiffeqG%)poltorecG%/ratpolytocoeffG%(rec*recG%(rec+ recG%,rectodiffeqG%,rectohomrecG%*rectoprocG%.seriestoalgeqG%/seriesto diffeqG%2seriestohypergeomG%-seriestolistG%0seriestoratpolyG%,seriesto recG%/seriestoseriesG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "More spe cifically, the " }{TEXT 264 4 "gfun" }{TEXT -1 74 " package will be us ed to prepare a system of PDE's for the application of " }{TEXT 263 5 "Mgfun" }{TEXT -1 55 " functions, and to solve the ODE that is output \+ by the " }{TEXT 265 5 "Mgfun" }{TEXT -1 9 " package." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 56 "Search for a System of PDE's Satisfied by the Integrand " }{XPPEDIT 18 0 "x*J[1](a*x)*I[1](a*x)*Y[0](x)*K[0](x) " "*,%\"xG\"\"\"-&%\"JG6#\"\"\"6#*&%\"aGF$F#F$F$-&%\"IG6#\"\"\"6#*&F,F $F#F$F$-&%\"YG6#\"\"!6#F#F$-&%\"KG6#F86#F#F$" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 11 "We use the " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 1 " " }{TEXT -1 98 "package to compute a system of PDE's satisfied \+ by each factor of the integrand. Next, we use the " }{HYPERLNK 17 "Mg fun" 2 "Mgfun" "" }{TEXT -1 2 " p" }{TEXT -1 62 "ackage to derive a sy stem of PDE's satisfied by their product." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "System of PDE's Satisfied by " }{XPPEDIT 18 0 "x" "I\"xG6 \"" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "The identity function " } {XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 54 " trivially satisfies the fo llowing differential system" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "sys1:=\{x*diff(h(x,a),x)-1,diff(h(x,a),a)\}:" ">%%sys1G<$,&*&%\"xG \"\"\"-%%diffG6$-%\"hG6$F'%\"aGF'F(F(\"\"\"!\"\"-F*6$-F-6$F'F/F/" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "where each entry " }{XPPEDIT 18 0 "expr" "I%exprG6\"" }{TEXT -1 33 " in the set denotes the equation " } {XPPEDIT 18 0 "expr=0" "/%%exprG\"\"!" }{TEXT -1 1 "." }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "System of PDE's Satisfied by " }{XPPEDIT 18 0 "J[1](a*x)" "-&%\"JG6#\"\"\"6#*&%\"aG\"\"\"%\"xGF*" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "We compute a system of PDE's for" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f:=BesselJ(1,a*x):" ">%\"fG-%( BesselJG6$\"\"\"*&%\"aG\"\"\"%\"xGF*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "by first computing an ODE with respect to the variable " } {XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 7 " using " }{HYPERLNK 17 "gfun [holexprtodiffeq]" 2 "gfun[holexprtodiffeq]" "" }{TEXT -1 1 "," } {TEXT -1 36 " and next considering symmetries of " }{XPPEDIT 18 0 "f" "I\"fG6\"" }{TEXT -1 29 " to derive a complete system." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "holexprtodiffeq(f,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%,(*&,&!\"\"\"\"\"*&%\"aG\"\"#%\"xGF+F(F(-% \"yG6#F,F(F(*&F,F(-%%diffG6$F-F,F(F(*&F,F+-F26$F1F,F(F(/-F.6#\"\"!F:/- -%\"DG6#F.F9,$F*#F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "de q:=op(remove(type,\",equation)):" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs(y(x)=h(x,a),deq);" "-%%subsG6$/-%\"yG6#%\"xG-%\"hG 6$F)%\"aG%$deqG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&!\"\"\"\"\"*& %\"aG\"\"#%\"xGF*F'F'-%\"hG6$F+F)F'F'*&F+F'-%%diffG6$F,F+F'F'*&F+F*-F1 6$F0F+F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs(\{x=a,a=x, y(x)=h(x,a)\},deq);" "-%%subsG6$<%/%\"xG%\"aG/F(F'/-%\"yG6#F'-%\"hG6$F 'F(%$deqG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&!\"\"\"\"\"*&%\"aG \"\"#%\"xGF*F'F'-%\"hG6$F+F)F'F'*&F)F'-%%diffG6$F,F)F'F'*&F)F*-F16$F0F )F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "x*diff(h(x,a),x)-a*d iff(h(x,a),a):" ",&*&%\"xG\"\"\"-%%diffG6$-%\"hG6$F$%\"aGF$F%F%*&F,F%- F'6$-F*6$F$F,F,F%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "s ys2:=\{\"\"\",\"\",\"\}:" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Sys tem of PDE's Satisfied by " }{XPPEDIT 18 0 "I[1](a*x)" "-&%\"IG6#\"\" \"6#*&%\"aG\"\"\"%\"xGF*" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "We com pute a system of PDE's for" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f:=BesselI(1,a*x):" ">%\"fG-%(BesselIG6$\"\"\"*&%\"aG\"\"\"%\"xGF*" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "in the same way as in the previo us section." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "holexprtodif feq(f,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%,(*&,&!\"\"\"\"\"*&% \"aG\"\"#%\"xGF+F'F(-%\"yG6#F,F(F(*&F,F(-%%diffG6$F-F,F(F(*&F,F+-F26$F 1F,F(F(/-F.6#\"\"!F:/--%\"DG6#F.F9,$F*#F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "deq:=op(remove(type,\",equation)):" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs(y(x)=h(x,a),deq);" "-%%subsG6$/- %\"yG6#%\"xG-%\"hG6$F)%\"aG%$deqG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, (*&,&!\"\"\"\"\"*&%\"aG\"\"#%\"xGF*F&F'-%\"hG6$F+F)F'F'*&F+F'-%%diffG6 $F,F+F'F'*&F+F*-F16$F0F+F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs(\{x=a,a=x,y(x)=h(x,a)\},deq);" "-%%subsG6$<%/%\"xG%\"aG/F(F'/- %\"yG6#F'-%\"hG6$F'F(%$deqG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&! \"\"\"\"\"*&%\"aG\"\"#%\"xGF*F&F'-%\"hG6$F+F)F'F'*&F)F'-%%diffG6$F,F)F 'F'*&F)F*-F16$F0F)F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "x*d iff(h(x,a),x)-a*diff(h(x,a),a):" ",&*&%\"xG\"\"\"-%%diffG6$-%\"hG6$F$% \"aGF$F%F%*&F,F%-F'6$-F*6$F$F,F,F%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sys3:=\{\"\"\",\"\",\"\}:" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 29 "System of PDE's Satisfied by " }{XPPEDIT 18 0 "Y[0](x) " "-&%\"YG6#\"\"!6#%\"xG" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "We com pute a system of PDE's for" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f:=BesselY(0,x):" ">%\"fG-%(BesselYG6$\"\"!%\"xG" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 55 "by first computing an ODE with respect to the vari able " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 10 " by using " } {HYPERLNK 17 "gfun[holexprtodiffeq]" 2 "gfun[holexprtodiffeq]" "" } {TEXT -1 25 ", and next encoding that " }{XPPEDIT 18 0 "f" "I\"fG6\"" }{TEXT -1 20 " does not depend on " }{XPPEDIT 18 0 "a" "I\"aG6\"" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "holexprtod iffeq(f,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"xG\"\"\"-%%di ffG6$-F(6$-%\"yG6#F%F%F%F&F&F*F&*&F%F&F,F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "sys4:=\{subs(y(x)=h(x,a),\"),diff(h(x,a),a)\};" "> %%sys4G<$-%%subsG6$/-%\"yG6#%\"xG-%\"hG6$F,%\"aG%\"\"G-%%diffG6$-F.6$F ,F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sys4G<$-%%diffG6$-%\"hG6$% \"xG%\"aGF-,(*&F,\"\"\"-F'6$-F'6$F)F,F,F0F0F3F0*&F,F0F)F0F0" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "System of PDE's Satisfied by " } {XPPEDIT 18 0 "K[0](x)" "-&%\"KG6#\"\"!6#%\"xG" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "We compute a system of PDE's for" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f:=BesselK(0,x):" ">%\"fG-%(BesselKG6$\"\"!%\"x G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "in the same way as in the pr evious section." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "holexprt odiffeq(f,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"xG\"\"\"-%% diffG6$-F(6$-%\"yG6#F%F%F%F&F&F*F&*&F%F&F,F&!\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {XPPEDIT 19 1 "sys5:=\{subs(y(x)=h(x,a),\"),diff(h(x,a),a)\}; " ">%%sys5G<$-%%subsG6$/-%\"yG6#%\"xG-%\"hG6$F,%\"aG%\"\"G-%%diffG6$-F .6$F,F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sys5G<$-%%diffG6$-%\"h G6$%\"xG%\"aGF-,(*&F,\"\"\"-F'6$-F'6$F)F,F,F0F0F3F0*&F,F0F)F0!\"\"" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "System of PDE's Satisfied by th e Product" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Computing a system fo r the product is now a simple call to " }{HYPERLNK 17 "Mgfun[`sys*sys` ]" 2 "Mgfun[`sys*sys`]" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "sys:=`sys*sys`(sys1,sys2,sys3,sys4,sys5);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$sysG<',D*&,(*&%\"aG\"\"%%\"xGF+!$_%\"%\"4 #\"\"\"*$F,F+\"$S(F/-%\"hG6$F,F*F/F/*(F*\"\"#F,F6-%%diffG6$-F86$-F86$- F86$F2F*F*F,F,F/\"$5#*&,(*$F,\"\"$!$3\"*&F*F+F,\"\"(\"$+(*$F,FG\"#?F/- F86$-F86$-F86$F2F,F,F,F/F/*&F,\"\"&-F86$-F86$FKF,F,F/!#)**(F*F/F,F+-F8 6$-F86$-F86$-F86$F>F,F,F,F,F/\"$S\"*&,(*$F,\"\"'F[o*$F,F6\"$0)*&F,F_oF *F+\"$?%F/FMF/F/*&F,F_o-F86$FSF,F/\"#N*&,(*&F*F/F,F/!%3>*&F*FRF,FR\"%+ G*&F*F/F,FR\"$+%F/FinF/F/*(F*F/F,FDFenF/!$q(*&F,F+FUF/\"$v\"*&,(*&F*F/ F,F+!%G:*&F*FRF,F+!%sKF*\"%77F/F>F/F/*&,(*$F,FR!$7%*&F*F+F,FRFHF,!%\"4 #F/FOF/F/*&F,FG-F86$FeoF,F/FR*&,(*$F*FD!#[*&F*FGF,F+!$+%*&F*FDF,F+F_pF /-F86$FF/!#E*&F,F/FOF/FD*&F*FDF[rF/!\")*(F*F/F,F6FgnF/!#7*&F *F6FF/F/*&,(F,!#@F]qFhtF_qF`uF/FOF/F/FetFftFgt\"#kFitFJF[u!#W*(F,FDF *F6-F86$F7F,F/!#5,B*&,(FDF/F)\"#7F0FJF/F2F/F/F5\"#;FcuFDFQFis*&,(F^oFi sF`o!\"#FboFJF/FMF/F/FdoF\\t*&,(F^pFftFjo!$!>F\\p!#CF/FinF/F/F`pF`uFbp F+*&,(Ffp!#;FhpFhvF*\"$Q#F/F>F/F/*&,(F,FhsF]qFbwF_qFhvF/FOF/F/FetF+Fgt \"#w*&,(Fir\"#)*F[sFiuFgrFftF/FFhs*&F*F6F[rF/Fis*&F*F/F " 0 "" {MPLTEXT 1 0 61 "ode:=op(int_of_sys(sys,x=-infinity..infinity,takayama_algo));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG,,*&%\"aG\"\"$-%\"hG6#F'\"\"\" \"#K*&,&*$F'\"\"(F,*$F'F(!\"\"F,-%%diffG6$-F56$-F56$-F56$F)F'F'F'F'F,F ,*&,&!\"$F,*$F'\"\"%\"$.\"F,F;F,F,*&,&*$F'\"\"#!\"%*$F'\"\"'\"#;F,F7F, F,*&,&*$F'\"\"&\"#tF'F(F,F9F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "However, the justification that the algorithm selected by the opti on " }{XPPEDIT 18 0 "takayama_algo" "I.takayama_algoG6\"" }{TEXT -1 0 "" }{TEXT -1 97 " applies to the integral under consideration is rathe r technical and is beyond this presentation." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Resolution of the Final ODE" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 11 "We use the " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 1 " " }{TEXT -1 34 "package to solve for the solution " }{XPPEDIT 18 0 "h(a)" "-%\"hG6#%\"aG" }{TEXT -1 199 " which corresponds to the i ntegral to be computed. Due to its integral representation, this func tion is analytic at 0, hence admits a Taylor expansion at 0. We proce ed to compute a closed form for " }{XPPEDIT 18 0 "h(a)" "-%\"hG6#%\"aG " }{TEXT -1 132 " by summation of this expansion. To this end, we det ermine a recurrence equation on the coefficients of the Taylor expansi on using " }{HYPERLNK 17 "gfun[diffeqtorec]" 2 "gfun[diffeqtorec]" "" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "ore:=diffeq torec(ode,h(a),u(n));" ">%$oreG-%,diffeqtorecG6%%$odeG-%\"hG6#%\"aG-% \"uG6#%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$oreG<%,&*&,&%\"nG\" \"\"\"\"#F*F*-%\"uG6#F)F*F**&,&F)!\"\"!\"'F*F*-F-6#,&F)F*\"\"%F*F*F*/- F-6#F*\"\"!/-F-6#\"\"$F:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 5 "Maple " }{TEXT -1 32 " readily solves this recurrence:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "rsol:=rsolve(ore,u(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%rsolG,**(,&-%\"uG6#\"\"##\"\"\"F+-F)6#\"\"!#F-\"\"%F --%&GAMMAG6#,&%\"nGF-F+F-F--F46#,&F7F-\"\"$F-!\"\"F+**,&F.F1F(#F F-),$FAF " 0 "" {MPLTEXT 1 0 44 "collect(map(normal,rsol,expanded),u,factor); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&**,&\"\"\"F&)-%'RootOfG6#,&*$%#_ ZG\"\"#F&F&F&%\"nGF&F&,&F&F&)!\"\"F/F&F&,&F/F&F.F&F2-%\"uG6#\"\"!F&#F& F.**,&F2F&F'F&F&F0F&F3F2-F56#F.F&F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "We perform the corresponding change of variable, " }{XPPEDIT 18 0 "n=2*p" "/%\"nG*&\"\"#\"\"\"%\"pGF&" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "subs(\{n=2*p,(-1)^n=1,RootOf(_Z^2+1 )^n=(-1)^p\},\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(,&\"\"\"F&)! \"\"%\"pGF&F&,&F)\"\"#F+F&F(-%\"uG6#\"\"!F&F&*(,&F(F&F'F&F&F*F(-F-6#F+ F&!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "so that " }{TEXT 275 5 " Maple" }{TEXT -1 27 " can sum the Taylor series:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sum(\"*a^(2*p),p=0..infinity);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$sumG6$*&,&*(,&\"\"\"F*)!\"\"%\"pGF*F*,&F-\"\" #F/F*F,-%\"uG6#\"\"!F*F**(,&F,F*F+F*F*F.F,-F16#F/F*!\"#F*)%\"aG,$F-F/F */F-;F3%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "h:=co llect(value(expand(\")),u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG, &*&,&*&%\"aG!\"#-%#lnG6#,&*$F)\"\"#!\"\"\"\"\"F2F2#F1F0*&F)F*-F,6#,&F/ F2F2F2F2#F2F0F2-%\"uG6#\"\"!F2F2*&,&F(F1F4F1F2-F:6#F0F2F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "It only remains to evaluate " }{XPPEDIT 18 0 "u[0]" "&%\"uG6#\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u[2]" "&%\"uG6#\"\"#" }{TEXT -1 20 ". We first compute " }{XPPEDIT 18 0 "u[ 0]" "&%\"uG6#\"\"!" }{TEXT -1 46 " and find it is 0 by inversion of li mits. Let" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f:=x*BesselJ(1, a*x)*BesselI(1,a*x)*BesselY(0,x)*BesselK(0,x):" ">%\"fG*,%\"xG\"\"\"-% (BesselJG6$\"\"\"*&%\"aGF&F%F&F&-%(BesselIG6$\"\"\"*&F,F&F%F&F&-%(Bess elYG6$\"\"!F%F&-%(BesselKG6$F5F%F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "be the integrand. We have:" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "int(limit(f,a=0),x=0..infinity);" "-%$intG6$-%&limitG6$ %\"fG/%\"aG\"\"!/%\"xG;F+%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "In the same way, each coefficient of the Taylor series for the integral is obtained by inve rsion of limits. In particular, " }{XPPEDIT 18 0 "kappa=u[2]" "/%&kap paG&%\"uG6#\"\"#" }{TEXT -1 6 ", but " }{TEXT 257 5 "Maple" }{TEXT -1 31 " is not capable of integrating:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "kappa=int(coeff(series(normal(diff(f,a,a)),a=0),a,0)/ 2,x=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&kappaG-%$intG6 $,$*(%\"xG\"\"$-%(BesselYG6$\"\"!F*\"\"\"-%(BesselKGF.F0#F0\"\"%/F*;F/ %)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "(This integral for " }{XPPEDIT 18 0 "kappa" "I&kappaG6\"" }{TEXT -1 33 " cannot be compu ted by a call to " }{TEXT 258 3 "int" }{TEXT -1 82 " using the Release 4, but the next release will probably be able to integrate it.)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "We obtain the following form for \+ " }{XPPEDIT 18 0 "h(a)" "-%\"hG6#%\"aG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "-combine(normal(-subs(\{u(0)=0,u(2) =kappa\},h)),ln,symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(-%#l nG6#*&,&*$%\"aG\"\"#!\"\"\"\"\"F.F.,&F*F.F.F.F.F.%&kappaGF.F+!\"#F-" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "It only remains to be proved tha t " }{XPPEDIT 18 0 "kappa=1/2/Pi" "/%&kappaG*(\"\"\"\"\"\"\"\"#!\"\"%# PiGF(" }{TEXT -1 300 ". We do not do it, since computing this last in tegral which is a constant lies outside the scope of the theory of hol onomy. With this example, we have reduced the problem of evaluating a parametrized integral to the evaluation of a non-parametrized integra l. In case there were no closed form for " }{XPPEDIT 18 0 "kappa" "I& kappaG6\"" }{TEXT -1 136 ", we could at least perform a simple numeric al evaluation and return a result in terms of this numerical value and the series above for " }{XPPEDIT 18 0 "kappa=1" "/%&kappaG\"\"\"" } {TEXT -1 1 "." }}}}}{MARK "0 1 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 } #%\"aG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "-combine(normal(-subs(\{u(0)=0,u(2) =kappa\},h)),ln,symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(-%#l nG6#*&,&*$%\"aG\"\"#!\"\"\"\"\"F.F.,&F*F.F.F.F.F.%&kappaGF.F+!\"#F-" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "It only remains to be proved tha t " }{XPPEDIT 18 0 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1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3 " 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 21 "Monomer-Dimer Tilings" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 25 "F. Ca zals, December 1997." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 247 "A fundamental problem \+ in lattice statistics is the monomer-dimer problem, in which the sites of a regular lattice are covered by non-overlapping monomers and dime rs, that is squares and pairs of neighbor squares. An example of such \+ a tiling for a " }{XPPEDIT 18 0 "mxn" "I$mxnG6\"" }{TEXT -1 17 " chess board with " }{XPPEDIT 18 0 "m=4" "/%\"mG\"\"%" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "n=5" "/%\"nG\"\"&" }{TEXT -1 119 " is depicted below. \+ The relative number of monomers and dimers can be arbitrary or may be \+ constrained to some density " }{XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 60 ", and the problem can be generalized to any fixed dimension " } {XPPEDIT 18 0 "d" "I\"dG6\"" }{TEXT -1 551 ". This model was introduce d long ago to investigate the properties of adsorbed diatomic molecule s on a crystal surface [Rob35], and its three-dimensional version occu rs in the theory of mixtures of molecules of different sizes [Gug52] a s well as the cell cluster theory of the liquid state [CoAl55]. Prac tically, most of the thermodynamic properties of these physical system s can be derived from the number of ways a given lattice can be cover ed, so that a considerable attention has been devoted to this counting question. For any fixed dimension " }{XPPEDIT 18 0 "d" "I\"dG6\"" } {TEXT -1 25 " and any monomer density " }{XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 4 ", a " }{TEXT 257 54 "provably good polynomial time approx imation algorithm " }{TEXT -1 93 "is exposed in [KenAl95]. But exact c ounting results are still unknown even in dimension two. " }}{PARA 13 "" 1 "" {INLPLOT "61-%'CURVESG6#7'7$\"\"!F(7$$\"\"\"F(F(7$F*$\"\"#F(7$ F(F-F'-F$6#7'F/F,7$F*$\"\"$F(7$F(F4F/-F$6#7'F6F37$F*$\"\"%F(7$F(F;F6-F $6#7'F)7$F-F(7$F-F*7$F*F*F)-F$6#7'FCFB7$F-F4F3FC-F$6#7'F37$F4F47$F4F;F :F3-F$6#7'FA7$F4F(7$F4F-7$F-F-FA-F$6#7'FRFQFKFGFR-F$6#7'FP7$F;F(7$F;F* 7$F4F*FP-F$6#7'FenFZ7$F;F-FQFen-F$6#7'FQFin7$F;F;FLFQ-F$6#7'F=7$F-F;7$ F-$\"\"&F(7$F(FcoF=-F$6#7'FaoF]o7$F;FcoFboFao-%'COLOURG6&%$RGBGF(F($\" *++++\"!\")-%(SCALINGG6#%,CONSTRAINEDG" 2 277 262 262 2 0 1 0 2 6 0 4 1 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12010 0 0 0 0 0 0 0 1 1 0 0 0 218 45 0 0 0 0 0 0 }}{PARA 0 "" 0 "" {TEXT -1 78 "The goal of this worksheet is to show that these question s are amenable to an " }{TEXT 263 36 "automated computer algebra treat ment" }{TEXT -1 311 " which goes from the specifications of the coveri ngs constructions in terms of Combstruct grammars, to the asymptotics \+ using rational generating functions and the numeric-symbolic method ex posed in [GoSa96]. In particular we shall be interested in enumerating the tilings for a vertical strip of constant width " }{XPPEDIT 18 0 " m" "I\"mG6\"" }{TEXT -1 193 " in terms of multivariate rational genera ting functions, from which the average number of pieces or the expecte d proportions of the three types of pieces in a random tiling are easi ly derived. " }}{PARA 0 "" 0 "" {TEXT -1 40 "This will also enable us \+ to establish a " }{TEXT 265 13 "provably good" }{TEXT -1 67 " sequence of upper and lower bounds for the connectivity constant " }{XPPEDIT 18 0 "tau=limit(g(n)^(1/n^2),n=infinity" "/%$tauG-%&limitG6$)-%\"gG6#% \"nG*&\"\"\"\"\"\"*$F+\"\"#!\"\"/F+%)infinityG" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "g(n)" "-%\"gG6#%\"nG" }{TEXT -1 38 " counts the number of ways to tile an " }{XPPEDIT 18 0 "nxn" "I$nxnG6\"" }{TEXT -1 12 " \+ cheesboard." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 "But before getting started, we need to load the Combstruct lib rary, as well as the piece of code doing the asymptotics of rational f ractions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "with(combstruc t): with(gfun): read `ratasympt.mpl`;read `./gfsolve.mpl`;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 10 "References" }}{PARA 0 "" 0 "" {TEXT -1 95 "[CoAl55] E.G.D. Cohen et al., A cell-cluster theory for the liquid state II, Physica XXI, 1955." }}{PARA 0 "" 0 "" {TEXT -1 240 "[Fin97] S. Finch, Favorite Mathematical Constants, http://www.mathsoft.com/cg i-shl/constant.bat.\n[GoSa96] X. Gourdon and B. Salvy, Effective Asymp totics of linear recurrences with rational coefficients, Discrete Math ematics, Vol. 153, 1996." }}{PARA 0 "" 0 "" {TEXT -1 316 "[Gug52] E.A. Guggenheim, Mixtures, Clarendon Press, 1952.\n[Ken95] C. Kenyon et al ., Approximating the number of Monomer-Dimer Coverings of a Lattice, P roc. of the 25th ACM STOC, 1993.\n[Rob35]J.K. Robert, Some properties \+ of adsorbed films of oxygen on tungsten, Proc. of the Royal Society of London, Vol. A 152, 1935." }}{PARA 0 "" 0 "" {TEXT -1 99 "[Sloa95] N. J.A. Sloane and S. Plouffe, The Encyclopedia of Integer Sequences, Aca demic Press, 1995." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 22 "A step-by- step example" }}{PARA 0 "" 0 "" {TEXT -1 33 "We first observe that the number " }{XPPEDIT 18 0 "T[n]" "&%\"TG6#%\"nG" }{TEXT -1 108 " counti ng the different tilings of a vertical slice of width 1 has a well kn own expression: since height " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 28 " can be reached from height " }{XPPEDIT 18 0 "n-1" ",&%\"nG\"\" \"\"\"\"!\"\"" }{TEXT -1 37 " by adding a monomer and from height " } {XPPEDIT 18 0 "n-2" ",&%\"nG\"\"\"\"\"#!\"\"" }{TEXT -1 33 " with a ve rtical dimer, we have " }{XPPEDIT 18 0 "T[n]=T[n-1]+T[n-2]" "/&%\"TG6 #%\"nG,&&F$6#,&F&\"\"\"\"\"\"!\"\"F+&F$6#,&F&F+\"\"#F-F+" }{TEXT -1 7 " with " }{XPPEDIT 18 0 "T[0]=1,T[1]=1" "6$/&%\"TG6#\"\"!\"\"\"/&F%6# \"\"\"\"\"\"" }{TEXT -1 82 ", that is the Fibonacci recurrence. This \+ can be checked directly with Combstruct:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "TGr:=\{T=Sequence(Union(monomer,dimer)),monomer=Z,dim er=Prod(Z,Z)\};" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$TGrG<%/%\"TG-%)S equenceG6#-%&UnionG6$%(monomerG%&dimerG/F.%\"ZG/F/-%%ProdG6$F1F1" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "And we can retrieve the correspond ing rational Generating Function with gfsolve:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "gfsolve(TGr,unlabelled, z);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<&/-%\"ZG6#%\"zGF(/-%(monomerGF'F(/-%&dimerGF'*$F(\" \"#/-%\"TGF',$*$,(!\"\"\"\"\"F(F8F/F8F7F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "More interesting is the case m=2 which we examine examine now." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 24 "Tiling a slice of width " }{XPPEDIT 18 0 "m=2" "/%\"mG\"\"#" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 442 "An example covering of a 2x6 lattice is depicted below. If we \+ draw a horizontal line at height 0, it turns out that we do not `cut' \+ any piece, which we encode by MM. At height 1, we cut the lefmost vert ical dimer but just touch the monomer topmost side, which we encode by PM. At height 2 the leftmost P turned into an M since we now touch t he dimer boundary, while on the right side we added a dimer and have \+ a P. More generally, we shall " }{TEXT 260 80 "assign to each height o f the construction containing a monomer or dimer boundary" }{TEXT -1 19 " a word of length " }{XPPEDIT 18 0 "m" "I\"mG6\"" }{TEXT -1 18 " \+ on the alphabet " }{XPPEDIT 18 0 "\{M,P" "<$%\"MG%\"PG" }{TEXT -1 18 " as follows: the " }{XPPEDIT 18 0 "i" "I\"iG6\"" }{TEXT -1 24 "th di git of the word is " }{XPPEDIT 18 0 "P" "I\"PG6\"" }{TEXT -1 89 " if a n horizontal line at this particular height splits a vertical domino l ocated in the " }{XPPEDIT 18 0 "i" "I\"iG6\"" }{TEXT -1 15 "th column, and " }{XPPEDIT 18 0 "M" "I\"MG6\"" }{TEXT -1 146 " otherwise. To sum marize our example we therefore have MM, PM, MP, MM, MM,MM at the heig hts 0,1,2,3,4,6. (BTW, M stands for Minus and P for Plus!)" }}{PARA 13 "" 1 "" {INLPLOT "6+-%'CURVESG6#7'7$\"\"!F(7$$\"\"\"F(F(7$F*$\"\"#F (7$F(F-F'-F$6#7'F)7$F-F(7$F-F*7$F*F*F)-F$6#7'F5F47$F-$\"\"$F(7$F*F:F5- F$6#7'F/F,F<7$F(F:F/-F$6#7'F@F97$F-$\"\"%F(7$F(FEF@-F$6#7'FG7$F*FE7$F* $\"\"'F(7$F(FMFG-F$6#7'FKFD7$F-FMFLFK-%(SCALINGG6#%,CONSTRAINEDG-%'COL OURG6&%$RGBGF(F($\"*++++\"!\")" 2 260 275 275 2 0 1 0 2 6 0 4 1 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12010 0 0 0 0 0 0 0 1 1 0 0 0 18 32 0 0 0 0 0 0 }}{PARA 0 "" 0 "" {TEXT -1 263 "This encoding is not one-to-one since whenever we find t wo consecutive Ms, we do not know wether they are on top of two monome rs or of a horizontal dimer. But it is sufficient to incrementally bu ild all the possible configurations by recording the status of the " } {TEXT 259 7 "fringe." }{TEXT -1 4 " If " }{XPPEDIT 18 0 "m=2" "/%\"mG \"\"#" }{TEXT -1 203 ", the possible fringes are MM,MP,PM and each of \+ them can be derived from a combination of the others and of monomers \+ and dimers. For example, the configuration MM can be reached in 5 dif ferent ways by:" }}{PARA 0 "" 0 "" {TEXT -1 111 " -stacking a hori zontal dimer H, two monomers C,C, or two vertical dimers V,V on top of a MM configuration," }}{PARA 0 "" 0 "" {TEXT -1 95 " -adding a mo nomer C to the right column of a PM configuration or to the left one o f a MP. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 295 "The remaining transitions follow similar rules. And in order to c haracterize the ordinate reached by the construction, we can mark the \+ height reached by the bottommost piece whose elevation gain is 1 or 2 \+ at each step of the construction. Putting everything together and asso ciating the symbols " }{XPPEDIT 18 0 "H,V,C" "6%%\"HG%\"VG%\"CG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "S" "I\"SG6\"" }{TEXT -1 118 " to th e number of horizontal dimers, vertical dimers, monomers and the heigh t yields the following Combstruct grammar:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 263 "Gr2:=\{MM=Union(Epsilon,Prod(S, MM, H), Prod(S, MM , C,C),\n Prod(S,PM, C),Prod(S,MP, C),Prod(S,S,MM,V,V)) ,\n PM=Union(Prod(S,MM, V, C), Prod(S,MP,V)),\n MP=Union(P rod(S,MM, C, V), Prod(S,PM,V)),\n H=Epsilon,V=Epsilon,C=Epsilon, S=Atom\}:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "The ordinary generat ing functions can be derived by Combstruct[gfsolve]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "GF2Sys:=gfsolve(Gr2, unlabelled, z, [[h,H ], [v,V], [c,C]]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'GF2SysG<)/-%# PMG6&%\"zG%\"hG%\"vG%\"cG**F*\"\"\"F,F/F-F/,2*&F*F/F,F/!\"\"F/F/*(F+F/ F*\"\"#F,F/F/*&F*F/F+F/F2*(F*F4F-F4F,F/F2*&F*F/F-F4F2*&F*\"\"$F,F9F/*& F*F4F,F4F2F2/-%#MPGF)F./-%#MMGF),$*&F0F2,&F1F/F2F/F/F2/-%\"CGF)F-/-%\" HGF)F+/-%\"VGF)F,/-%\"SGF)F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "F urthermore we can isolate the GF corresponding to the MM fringes; the \+ coefficient of " }{XPPEDIT 18 0 "z^n*h^i*v^j*c^l" "**)%\"zG%\"nG\"\" \")%\"hG%\"iGF&)%\"vG%\"jGF&)%\"cG%\"lGF&" }{TEXT -1 59 " in this GF c ounts the number of ways to tile a chessboard " }{XPPEDIT 18 0 "2xn" " *&\"\"#\"\"\"%#xnGF$" }{TEXT -1 19 " with respectively " }{XPPEDIT 18 0 "j,k" "6$%\"jG%\"kG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "l" "I\"lG6 \"" }{TEXT -1 45 " horizontal and vertical dimers and monomers:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "GF2:=subs(GF2Sys,MM(z,h,v,c) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$GF2G,$*&,2*&%\"zG\"\"\"%\"vGF *!\"\"F*F**(%\"hGF*F)\"\"#F+F*F**&F)F*F.F*F,*(F)F/%\"cGF/F+F*F,*&F)F*F 2F/F,*&F)\"\"$F+F5F**&F)F/F+F/F,F,,&F(F*F,F*F*F," }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 193 "The number of configurations up to a given height independently of the number and kind of pieces used can be retrieved \+ by erasing the dimers and monomers markers followed by a Taylor expans ion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "GF2h:=subs([h=1,v=1 ,c=1],GF2);series(GF2h,z=0,11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%% GF2hG,$*&,*%\"zG!\"$\"\"\"F**$F(\"\"#!\"\"*$F(\"\"$F*F-,&F(F*F-F*F*F- " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+;%\"zG\"\"\"\"\"!\"\"#\"\"\"\"\"( \"\"#\"#A\"\"$\"#r\"\"%\"$G#\"\"&\"$L(\"\"'\"%cB\"\"(\"%tv\"\")\"&UV# \"\"*\"&V#y\"#5-%\"OG6#F%\"#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 147 "This sequence does not appear in [Sloa95]. It can be checked that the se values match those computed directly from the grammer by Combstruct [count]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(count([MM,G r2], size=i), i=0..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-\"\"\"\"\"# \"\"(\"#A\"#r\"$G#\"$L(\"%cB\"%tv\"&UV#\"&V#y" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Another way to compute the exact number of tilings for large values of " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 107 " is th rough the recurrence equation satisfied by the Taylor coefficients and computed by gfun[diffeqtorec]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "diffeqtorec(y(z)-GF2h,y(z),u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&,*-%\"uG6#%\"nG\"\"\"-F&6#,&F(F)F)F)!\"\"-F&6#,&F(F) \"\"#F)!\"$-F&6#,&F(F)\"\"$F)F)/-F&6#\"\"!F)/-F&6#F)F1/-F&6#F1\"\"(" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "p2:=rectoproc(\",u(n)):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "for i from 1 to 10 do i,p2 (i) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"#\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\" $\"#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"%\"#r" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$\"\"&\"$G#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"'\" $L(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"(\"%cB" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\")\"%tv" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"*\"&U V#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#5\"&V#y" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 12 "For example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "p2(1000);evalf(\");" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#\"fjl&o^*pV?W$>sV#>V'yw`\\P[MD(f-:;pr2Z$3HZ&*z]s<]sr264FFcnLNvwX*QD; yN&*y.r)*>PUH\"z3d!RbB.kSNt=ioC$[,km?x$>`z\"oByy%fuy,(GIi(36(G&G=Mh$\\[[xofF^^^f2B6 /C&p!p,2cTm!))e\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+k1))e\")\"$( \\" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "But as we shall see now, as ymptotic estimates can be derived much faster." }}}}{SECT 0 {PARA 4 " " 0 "" {TEXT -1 45 "Asymptotic estimates of the number of tilings" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 245 "We have just seen that the number of configurations is encoded by the rational generating function GF2h (z). An elegant way to access its Taylor coefficients is therefore thr ough a full partial fraction decomposition yielding linear denominator s:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "fpf:=convert(GF2h,ful lparfrac,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fpfG-%$SumG6$*&,(%' _alphaG#!\")\"#P*$F*\"\"##\"\"(\"#u#!#6F2\"\"\"F5,&%\"zGF5F*!\"\"F8/F* -%'RootOfG6#,*%#_ZG!\"$F5F5*$F>F/F8*$F>\"\"$F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "The term in " }{XPPEDIT 18 0 "z^n" ")%\"zG%\"nG" } {TEXT -1 47 " comes from the contributions of the roots of " } {XPPEDIT 18 0 "Z^3-Z^2-3*Z+1=0" "/,**$%\"ZG\"\"$\"\"\"*$F%\"\"#!\"\"*& \"\"$F'F%F'F*\"\"\"F'\"\"!" }{TEXT -1 20 " in the expansion of" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "el:=op(1,fpf);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#elG*&,(%'_alphaG#!\")\"#P*$F'\"\"##\"\"(\"#u# !#6F/\"\"\"F2,&%\"zGF2F'!\"\"F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "and since there are 3 singularities, the main asymptotic contribu tion comes from the one with smallest modulus:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "fsolve(-3*_Z+1+_Z^3-_Z^2,_Z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$!+/V>\"[\"!\"*$\"+v\"y56$!#5$\"+(['3q@F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "root1:=RootOf(-3*_Z+1+_Z^3- _Z^2,.3111078175);\nroot2:=RootOf(-3*_Z+1+_Z^3-_Z^2,-1.481194304);\nro ot3:=RootOf(-3*_Z+1+_Z^3-_Z^2,2.170086487); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&root1G-%'RootOfG6$,*%#_ZG!\"$\"\"\"F+*$F)\"\"#!\"\"* $F)\"\"$F+$\"+v\"y56$!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&root2G- %'RootOfG6$,*%#_ZG!\"$\"\"\"F+*$F)\"\"#!\"\"*$F)\"\"$F+$!+/V>\"[\"!\"* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&root3G-%'RootOfG6$,*%#_ZG!\"$\" \"\"F+*$F)\"\"#!\"\"*$F)\"\"$F+$\"+(['3q@!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "On this example the dominant pole is clearly " } {XPPEDIT 18 0 "0.31" "$\"#J!\"#" }{TEXT -1 45 " so that the main contr ibution is encoded by:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "e l1:=subs(_alpha=root1,el);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$el1G* &,(-%'RootOfG6$,*%#_ZG!\"$\"\"\"F-*$F+\"\"#!\"\"*$F+\"\"$F-$\"+v\"y56$ !#5#!\")\"#P*$F'F/#\"\"(\"#u#!#6F " 0 "" {MPLTEXT 1 0 9 "evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$,&%\"zG\"\"\"$!+v\"y56$!#5F'!\"\"$!+^dfn?F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Extracting the term in " } {XPPEDIT 18 0 "z^n" ")%\"zG%\"nG" }{TEXT -1 50 " in the previous expre ssion produces the estimate:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "es2:=n->.2067595751*(1/.3111078175)^(n+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$es2G:6#%\"nG6\"6$%)operatorG%&arrowGF(,$)$\"+V(>V@$! \"*,&9$\"\"\"F3F3$\"+^dfn?!#5F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "seq(es2(i),i=1..10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6,$\"+3#4i8#!\"*$\"+J%fk'oF%$\"+7'*42A!\")$\"+bQK%4(F*$\"+BDM!G#!\"($ \"+#**\\(HtF/$\"+6g,cB!\"'$\"+!y))Hd(F4$\"+O2?MC!\"&$\"+\"[*HCyF9" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "To sum up, from the rational gener ating function we have:" }}{PARA 0 "" 0 "" {TEXT -1 49 "-performed a f ull partial fraction decomposition," }}{PARA 0 "" 0 "" {TEXT -1 65 "-c omputed the singularities and sorted them by increasing moduli," }} {PARA 0 "" 0 "" {TEXT -1 69 "-extracted the contribution of the singul arity with smallest modulus." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 227 "The key step consists in deciding which are th e singularity (ies) with smallest modulus (i), and can be performed nu merically using properties of polynomials with integer coefficients -- see [GoSa96]. This is implemented by the " }{TEXT 261 9 "ratasympt" } {TEXT -1 27 " function --whose optional " }{XPPEDIT 18 0 "4" "\"\"%" } {TEXT -1 157 "th argument corresponds to the number of singularity lay ers the user wants to take into account. In particular to retrieve the main contribution, one writes:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "layer1:=ratasympt(GF2h,z,n,1);nbCfs1:=evalf(layer1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'layer1G,$*&,(-%'RootOfG6$,*%#_ZG!\"$\"\" \"F.*$F,\"\"#!\"\"*$F,\"\"$F.$\"-mu\"y56$!#7#!\")\"#P*$F(F0#\"\"(\"#u# !#6F=F.F.)F(,&%\"nGF.F.F.F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'nb Cfs1G,$*$)$\"+v\"y56$!#5,&%\"nG\"\"\"$F-\"\"!F-!\"\"$\"+^dfn?F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "And to take into account all the l ayers:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "layers:=ratasympt (GF2h,z,n);nbCfs:=evalf(layers);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% 'layersG,(*&,(-%'RootOfG6$,*%#_ZG!\"$\"\"\"F.*$F,\"\"#!\"\"*$F,\"\"$F. $\"-mu\"y56$!#7#!\")\"#P*$F(F0#\"\"(\"#u#!#6F=F.F.)F(,&%\"nGF.F.F.F1F1 *&,(-F)6$F+$!-4/V>\"[\"F?F7*$FEF0F;F>F.F.)FEFAF1F1*&,(-F)6$F+$\"-j'['3 q@F?F7*$FMF0F;F>F.F.)FMFAF1F1" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&nb CfsG,(*$)$\"+v\"y56$!#5,&%\"nG\"\"\"$F-\"\"!F-!\"\"$\"+^dfn?F**$)$!+/V >\"[\"!\"*F+F0$!+$>T9z$F**$)$\"+(['3q@F7F+F0$\"+Sa%Qs\"F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "We can check that the second approximatio n is more accurate:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "eval f(seq(subs(n=i, layer1), i=1..10));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 ,$\"+3#4i8#!\"*$\"+I%fk'oF%$\"+6'*42A!\")$\"+cQK%4(F*$\"+BDM!G#!\"($\" +#**\\(HtF/$\"+7g,cB!\"'$\"+\"y))Hd(F4$\"+N2?MC!\"&$\"+#[*HCyF9" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "evalf(seq(subs(n=i, layers), i=1..10));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6,$\"+++++?!\"*$\"+&***** **pF%$\"+)******>#!\")$\"+&******4(F*$\"+)*****zA!\"($\"+\"*****HtF/$ \"+(****fN#!\"'$\"+\"****Hd(F4$\"+'***>MC!\"&$\"+*)**HCyF9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "seq(p2(i),i=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"#\"\"(\"#A\"#r\"$G#\"$L(\"%cB\"%tv\"&UV#\"&V#y " }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 37 "The proportion of monomers \+ and dimers" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "We now address the \+ computation of the average number of pieces in a random tiling. From t he multivariate generating function " }{XPPEDIT 18 0 "GF2(z,h,v,c)" "- %$GF2G6&%\"zG%\"hG%\"vG%\"cG" }{TEXT -1 51 " we can merge the three ty pes of pieces as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "GF2;stij:=subs([h=t,v=t,c=t], GF2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,2*&%\"zG\"\"\"%\"vGF(!\"\"F(F(*(%\"hGF(F'\"\"#F)F(F(*&F'F(F ,F(F**(F'F-%\"cGF-F)F(F**&F'F(F0F-F**&F'\"\"$F)F3F(*&F'F-F)F-F*F*,&F&F (F*F(F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%stijG,$*&,,*&%\"zG\"\" \"%\"tGF*!\"#F*F**&F)\"\"#F+\"\"$!\"\"*&F)F*F+F.F0*&F)F/F+F/F*F0,&F(F* F0F*F*F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The coefficient of \+ " }{XPPEDIT 18 0 "z^i*t^j" "*&)%\"zG%\"iG\"\"\")%\"tG%\"jGF&" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "stij" "I%stijG6\"" }{TEXT -1 40 " counts t he number of tilings at height " }{XPPEDIT 18 0 "i" "I\"iG6\"" }{TEXT -1 14 " with exactly " }{XPPEDIT 18 0 "j" "I\"jG6\"" }{TEXT -1 110 " p ieces of any type. To get the total number of pieces we just have to c ompute the derivative with respect to " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 16 " and substitute " }{XPPEDIT 18 0 "t=1" "/%\"tG\"\"\"" } {TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sstij:=sub s(t=1, diff(stij,t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sstijG,&*( ,*%\"zG!\"$\"\"\"F**$F(\"\"#!\"\"*$F(\"\"$F*!\"#,&F(F*F-F*F*,(F(!\"%F+ F)F.F/F*F**&F'F-F(F*F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "For ex ample, the total number of dimers and monomers used in all the configu rations tilling the square " }{XPPEDIT 18 0 "2x2" "*&\"\"#\"\"\"%#x2GF $" }{TEXT -1 7 " is 20:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " series(\",z=0,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+-%\"zG\"\"$\"\" \"\"#?\"\"#\"#%*\"\"$\"$-%\"\"%-%\"OG6#\"\"\"\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "As before, we can compute an estimate of the tot al number of pieces in all the configurations at a given height:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ratasympt(sstij,z,n,1);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#,(*(,(*$-%'RootOfG6$,*%#_ZG!\"$\"\"\"F -*$F+\"\"#!\"\"*$F+\"\"$F-$\"-mu\"y56$!#7F/#\"\"(\"#u#\"\"&\"#PF-F'#!# 8F8F-,&%\"nGF-F-F-F-)F',&F?F-F/F-F0F-*&,(F'#!$w$\"%p8F&#\"$^&\"%QF#!%N 5FIF-F-)F'F>F0F0*&,(F'#\"#RF8F&#F5F;#\"#VF8F-F-FLF0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "nBDPiecesN:=evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+nBDPiecesNG,&*&,&%\"nG\"\"\"$F)\"\"!F)F))$\" +v\"y56$!#5,&F(F)$\"\"#F+F)!\"\"$\"+l(oO'*)!#6*$)F-F'F3$!+O`q'p#F/" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "So that the average number of pie ces is asymptotically equivalent to:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "avNbD:=expand(nBDPiecesN/nbCfs1);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&avNbDG,&%\"nG$\"+)G2NR\"!\"*$\"+d6iB*)!#6\"\"\"" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "And the average number of pieces per layer in a tiling of height " }{XPPEDIT 18 0 "n" "I\"nG6\"" } {TEXT -1 14 " is therefore:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "asympt(\"/n,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&$\"+)G2NR\"! \"*\"\"\"*$%\"nG!\"\"$\"+d6iB*)!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 142 "The number of occurrences and the proportions of dimers and mo nomers can be computed in the same way by erasing the irrelevant indet erminates:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 270 "pieceProport ion:=proc(MGF, keptPiece)\n local forSubs, stij, sstij, nbp;\n\n forSu bs:=\{h=1,v=1,c=1\} minus \{keptPiece=1\}; \n stij:=subs([op(forSubs)] , MGF);\n sstij:=subs(keptPiece=1, diff(stij,keptPiece));\n nbp:=evalf (ratasympt(sstij,z,n,1));\n asympt(nbp/nBDPiecesN,n,2)\nend:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "And we end up with:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "nbh:=pieceProportion(GF2,h);\nnbv:= pieceProportion(GF2,v);\nnbc:=pieceProportion(GF2,c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$nbhG,&$\"+b^t$[\"!#5\"\"\"-%\"OG6#*$%\"nG!\"\"F )" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$nbvG,&$\"+s*R&oG!#5\"\"\"-%\"O G6#*$%\"nG!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$nbcG,&$\"+%)[s Zc!#5\"\"\"-%\"OG6#*$%\"nG!\"\"F)" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 25 "Plotting routines archive" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "The figures above were plotted with the following functions:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 203 "dominoH:=proc(x,y) [[x,y], \+ [x+2,y], [x+2,y+1], [x,y+1], [x,y]] end:\ndominoV:=proc(x,y) [[x,y], [ x+1,y], [x+1,y+2], [x,y+2], [x,y]] end:\ndominoC:=proc(x,y) [[x,y], [x +1,y], [x+1,y+1], [x,y+1], [x,y]] end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 238 "plot([dominoV(0,0), dominoC(0,2),dominoC(0,3),\n \+ dominoC(1,0),dominoV(1,1),dominoH(1,3),\n dominoV(2,0),dominoC( 2,2), \n dominoC(3,0),dominoC(3,1),dominoV(3,2),\n dominoH(0 ,4),dominoH(2,4)],scaling=constrained,color=blue);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "plot([dominoV(0,0), dominoC(1,0),dominoV(1,1),dominoC(0,2),domino H(0,3),dominoV(0,4),dominoV(1,4)], scaling=constrained,color=blue);" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 39 "Automatic counting in a slice of width " }{XPPEDIT 18 0 "m" "I\"mG 6\"" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 34 "Computing the generating f unctions" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "We now show how to aut omate the previous computations for any integer " }{XPPEDIT 18 0 "m" " I\"mG6\"" }{TEXT -1 44 ". The first task consists in generating the " }{XPPEDIT 18 0 "2^m-1" ",&)\"\"#%\"mG\"\"\"\"\"\"!\"\"" }{TEXT -1 30 " words on the binary alphabet " }{XPPEDIT 18 0 "\{M,P\}" "<$%\"MG%\"PG " }{TEXT -1 63 ", and this is easily done with a Combstruct grammar as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 273 "allMPWords:=p roc(m::integer)\n local i, MPGr, mps1, mps2, Pm;\n\n MPGr:=\{AllMP=Seq uence(MP), MP=Union(M,P), M=Atom, P=Atom\};\n mps1:=allstructs([AllMP, MPGr], size=m);\n mps2:=convert(map(proc(x) cat(op(x)) end, mps1), se t);\n Pm:=cat(seq(P,i=1..m)); \n [op(mps2 minus \{Pm\})]\nend:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "For example if " }{XPPEDIT 18 0 "m =3" "/%\"mG\"\"$" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "allMPWords(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)% $PMMG%$MMPG%$MMMG%$PPMG%$MPPG%$MPMG%$PMPG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "More interesting is the generation of the transitions bet ween these words. Let " }{XPPEDIT 18 0 "pattern" "I(patternG6\"" } {TEXT -1 66 " be one of them and suppose we want to figure out all the fringes " }{XPPEDIT 18 0 "pattern" "I(patternG6\"" }{TEXT -1 46 " can be derived from. Suppose for example the " }{XPPEDIT 18 0 "i" "I\"iG6 \"" }{TEXT -1 13 "th letter of " }{XPPEDIT 18 0 "pattern" "I(patternG6 \"" }{TEXT -1 6 " is a " }{XPPEDIT 18 0 "P" "I\"PG6\"" }{TEXT -1 22 "; this means that the " }{XPPEDIT 18 0 "i" "I\"iG6\"" }{TEXT -1 24 "th \+ letter of the fringe " }{XPPEDIT 18 0 "pattern" "I(patternG6\"" } {TEXT -1 22 " was derived from was " }{XPPEDIT 18 0 "M" "I\"MG6\"" } {TEXT -1 50 " and that a vertical dimer was put on top of this " } {XPPEDIT 18 0 "M" "I\"MG6\"" }{TEXT -1 31 ". Similar rules applies if \+ the " }{XPPEDIT 18 0 "i" "I\"iG6\"" }{TEXT -1 14 "th digit is a " } {XPPEDIT 18 0 "M" "I\"MG6\"" }{TEXT -1 86 ". And since the letter of a given fringe are independent --except for two consecutive " } {XPPEDIT 18 0 "M" "I\"MG6\"" }{TEXT -1 118 "s that may come from an ho rizontal dimer, it suffices to recursively examine the digits from lef t to right as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1186 "#--pattern is the fringe to be built, e.g. MMPMM\nrecComesFrom:=proc( pattern::string, idx::integer, prefix::string, mul::list, result::tabl e)\n local prodRes, m, Mm, Pm;\n \n if (idx>length(pattern)) then #--s tores the result into an indexed table\n prodRes:=Prod(S,prefix, op(m ul)); \n if not assigned(result[pattern]) then result[pattern]:=\{pr odRes\}\n else result[pattern]:=resu lt[pattern] union \{prodRes\}\n fi\n else\n #--we examine the idx^\{ th\} letter of the target\n if substring(pattern,idx)=P then\n recC omesFrom(pattern, idx+1, cat(prefix,M), [op(mul), V], result) \n else #target=M\n recComesFrom(pattern, idx+1, cat(prefix,P), mul, result );\n recComesFrom(pattern, idx+1, cat(prefix,M), [op(mul), C], resul t);\n \n #--we may have MM=Prod(MM,H)\n if (length(pattern)>idx ) and (substring(pattern,idx+1)=M) then\n recComesFrom(pattern, idx +2, cat(prefix,M,M), [op(mul), H], result)\n fi\n fi\n fi;\n\n #--s ome extra work for M^m\n m:=length(pattern);\n Mm:=cat(seq(M,i=1..m)); \n if pattern=Mm then\n Pm:=cat(seq(P,i=1..m));\n result[Mm]:=result [Mm] minus \{Prod(S,Pm)\} \n union \{Epsilon,P rod(S,S,Mm,seq(V,i=1..m))\}\n fi\nend: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Here is the table for " }{XPPEDIT 18 0 "m=3" "/%\"mG\"\"$ " }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "table3: =table():for i in allMPWords(3) do recComesFrom(i, 1, ``, [], table3) \+ od:print(table3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7)/%$M PMG<&-%%ProdG6%%\"SG%$PMPG%\"VG-F+6'F-%$MMMG%\"CGF/F3-F+6&F-%$MMPGF3F/ -F+6&F-%$PMMGF/F3/F2 " 0 "" {MPLTEXT 1 0 242 "setGrammarFromTable:=proc(aTable)\n local aList, transitions, x; \n aList:=op(op(aTable));#--[a=\{Prod(...), Prod(...)\}, ...]\n transi tions:=seq(op(1,x)=Union(op(op(2,x))), x=aList);\n \{transitions\} uni on \{H=Epsilon,V=Epsilon,C=Epsilon,S=Atom\}\nend:" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 24 "This yields the grammar:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Gr3:=setGrammarFromTable(table3);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%$Gr3G<-/%$MMPG-%&UnionG6'-%%ProdG6&%\"SG%$PMMG %\"CG%\"VG-F,6&F.%$MPMGF0F1-F,6'F.%$MMMGF0F0F1-F,6&F.F7%\"HGF1-F,6%F.% $PPMGF1/F/-F)6'-F,6%F.%$MPPGF1-F,6&F.F4F1F0-F,6'F.F7F1F0F0-F,6&F.F7F1F :-F,6&F.F'F1F0/FC-F)6$-F,6'F.F7F0F1F1-F,6&F.F/F1F1/%$PMPG-F)6$-F,6'F.F 7F1F0F1-F,6&F.F4F1F1/F=-F)6$-F,6'F.F7F1F1F0-F,6&F.F'F1F1/F4-F)6&-F,6%F .FTF1-F,6'F.F7F0F1F0-F,6&F.F'F0F1-F,6&F.F/F1F0/F7-F)6/%(EpsilonG-F,6(F .F.F7F1F1F1-F,6%F.F=F0-F,6&F.F/F0F0-F,6%F.FTF0-F,6&F.F4F0F0-F,6%F.FCF0 -F,6%F.F/F:-F,6&F.F'F0F0-F,6'F.F7F0F0F0-F,6&F.F7F0F:-F,6%F.F'F:-F,6&F. F7F:F0/F:Fjo/F1Fjo/F0Fjo/F.%%AtomG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "This is solved as usual:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "MM3GFSys:=gfsolve(Gr3, unlabelled, z, [[h,H], [v,V], [c,C]]); \nMM3GF:=subs(MM3GFSys,MMM(z,h,v,c));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)MM3GFSysG<-/-%$PMMG6&%\"zG%\"hG%\"vG%\"cG**F*\"\"\"F,F/,.*(F*F/F -F/F,\"\"#!\"\"*$F-F2F3F+F3*(F*\"\"$F,\"\"&F-F/F/*(F*F2F,F6F+F/F/*(F*F /F-F6F,F/F3F/,LF3F/*&F*F2F,F6F6*(F*F2F-F2F,F2F7*&F*\"\"%F,\"\"'!\"$*(F *F/F-F/F,F/F/*(F*F6F,F>F-F/!\"#*(F*F>F-F>F,F>FC*(F*F/F-F/F+F/F2*(F*F6F -F6F,F6F/**F*F>F,F>F+F/F-F2F2*(F*F7F,F?F- F6F/*(F-F2F*F>F,F7!\"&**F*F7F ,F?F-F/F+F/FC*(F-F>F*F2F,F/F2*(F*F7F-F/F,\"\"(F/*(F*F>F,F>F+F2FC**F-F2 F*F2F+F/F,F/F2*&F*F?F,\"\"*F/*&F*F/F-F6F/*(F*F2F+F2F,F/F2*(F*F6F-F7F,F 2F/F3/-%\"CGF)F-/-%\"HGF)F+/-%\"SGF)F*/-%\"VGF)F,/-%$PMPGF),$*,F*F/F-F /F,F2,,F/F/F;F3FBF/FF-F2F/*(F*F2F-F6F,F2F3*(F*F/F+F/F,F/F/* (F*F6F,F>F+F/F3F/F:F3F3/-%$PPMGF)Ffo/-%$MMMGF),$*&,.F/F/F;FCFAF3F=F/FB F/F%&MM3GFG,$*&,.\"\"\"F(*&%\"zG\"\"#%\"vG\"\"$!\"#*(F*F(%\"cGF(F, F(!\"\"*&F*\"\"%F,\"\"'F(*(F*F-F,F3F0F(F(*(F*F+F0F+F,F+F.F(,LF1F(F)F-F 6\"\"&F2!\"$F/F(F5F.*(F*F3F0F3F,F3F.*(F*F(F0F(%\"hGF(F+*(F*F-F0F-F,F-F (**F*F3F,F3F " 0 "" {MPLTEXT 1 0 356 "getGrammar:=proc(m::integer)\n local i, MP Table;\n \nMPTable:=table();\nfor i in allMPWords(m) do recComesFrom(i ,1,``,[],MPTable) od;\nsetGrammarFromTable(MPTable)\nend:\n\ngetMmGFun :=proc(m::integer)\n local i, MPTable,Grm,MMmGFSys;\n\nGrm:=getGrammar (m);\nMMmGFSys:=gfsolve(Grm, unlabelled, z, [[h,H], [v,V], [c,C]]);\ns ubs(MMmGFSys,cat(seq(M,i=1..m))(z,h,v,c));\nend:" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 182 "The computation to be carried out being quite hea vy for 4-variate generating functions, we can alleviate it be keeping \+ only the markers for the total number of pieces and the height:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 263 "getMmGFunZ:=proc(m::integer )\n local i, MPTable,Grm,GrmM,MMmGFSys;\n\nMPTable:=table();\nfor i in allMPWords(m) do recCome sFrom(i,1,``,[],MPTable) od;\nGrm:=setGrammar FromTable(MPTable);\nMMmGFSys:=gfsolve(Grm, unlabelled, z);\nsubs(MMmG FSys,cat(seq(M,i=1..m))(z));\nend:" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 11 "Asymptotics" }}{EXCHG {PARA 12 "" 1 "" {TEXT -1 64 "We can now \+ compute the generating functions for small values of " }{XPPEDIT 18 0 "m" "I\"mG6\"" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "gf:='gf':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "for i fro m 1 to 5 do i,time(assign(gf[i],getMmGFunZ(i))),gf[i] od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\"$\"$`&!\"$,$*$,(!\"\"F#*$%\"zG\"\"#F#F ,F#F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"#$\"$a&!\"$,$*&,*%\"zGF &\"\"\"F+*$F*F#!\"\"*$F*\"\"$F+F-,&F*F+F-F+F+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"$$\"%$f$!\"$,$*&,,*$%\"zG\"\"%\"\"\"*$F+F#F-*$F+\" \"#!\"%F+!\"\"F-F-F-,,F/\"#9F2F-F+F,*$F+\"\"'F-F*!#5F2F2" }}{PARA 11 " " 1 "" {XPPMATH 20 "6%\"\"%$\"&Eg\"!\"$,$*&,2\"\"\"F*%\"zG!\"%*$F+\"\" #!#:*$F+\"\"$\"#?*$F+\"\"(F**$F+\"\"&!#6*$F+\"\"'!\"#* $F+F#\"#5F*,6*$F +\"\"*F**$F+\"\")!\"\"F3!#BF8\"#HF5\"#\"*F;!$6\"F0!#TF-\"#TF+F?FBF*FBF B" }}{PARA 12 "" 1 "" {XPPMATH 20 "6%\"\"&$\"&tz)!\"$,$*&,H*$%\"zG\"#= \"\"\"*$F+\"#<\"\"#*$F+\"#;!#X*$F+\"#:!#o*$F+\"#9\"$a'*$F+\"#8\"$q)*$F +\"#7!%?Q*$F+\"#6!%+Z*$F+\"#5\"%b#**$F+\"\"*\"%[%**$F+\"\")!&v6\"*$F+ \"\"(!%Kv*$F+\"\"'\"%cp*$F+F#\"%%*>*$F+\"\"%!%%z\"*$F+\"\"$!#))*$F+F0 \"$8\"F+FP!\"\"F-F-,L*$F+\"#?F-*$F+\"#>F0F*!#lF.!$S\"F1\"%\"G\"F4\"%QD F7!&m.\"F:!&/w\"F=\"&`&QF@\"&e,&FC!&BO(FF!&#[gFI\"&lY(FL\"&kl#FO!&1^$F R!$)*)FT\"%dZFWF2FZ!$H#F+!#9F-F-FfnFfn" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "For bigger ones, the grammar size, that is " }{XPPEDIT 18 0 "2^m-1" ",&)\"\"#%\"mG\"\"\"\"\"\"!\"\"" }{TEXT -1 36 ", inherent ly yields a linear system " }{XPPEDIT 18 0 "(2^m-1)*x*(2^m-1)" "*(,&) \"\"#%\"mG\"\"\"\"\"\"!\"\"F'%\"xGF',&)\"\"#F&F'\"\"\"F)F'" }{TEXT -1 84 " with large coefficients whose resolution is very much time consum ing. So that for " }{XPPEDIT 18 0 "m >=6" "1\"\"'%\"mG" }{TEXT -1 95 ", a better altern ative to running getMmGFunZ(m) is to retrieve the re sult in the archive below!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "Fr om these generating functions we can easily isolate the main contribut ion to the asymptotic equivalent with the ratasympt procedure:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "asGf:='asGf':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "for i from 1 to 6 do assign(asGf[i] ,ratasympt(gf[i],z,n,1)),evalf(asGf[i]) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)$\"+())R.='!#5,&%\"nG\"\"\"$F+\"\"!F+!\"\"$\"+bf8s WF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)$\"+v\"y56$!#5,&%\"nG\"\" \"$F+\"\"!F+!\"\"$\"+^dfn?F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)$ \"+/$p(4;!#5,&%\"nG\"\"\"$F+\"\"!F+!\"\"$\"+cYAu))!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)$\"+>Y\\#H)!#6,&%\"nG\"\"\"$F+\"\"!F+!\"\"$\" +k[%*=RF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)$\"+i-NuU!#6,&%\"nG \"\"\"$F+\"\"!F+!\"\"$\"+*pr]r\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,$*$)$\"+bP+.A!#6,&%\"nG\"\"\"$F+\"\"!F+!\"\"$\"+,oREv!#7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "It should be observed that these estimate s correspond to huge expressions. For " }{XPPEDIT 18 0 "m=5" "/%\"mG\" \"&" }{TEXT -1 13 " for example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "asGf[5];" }}{PARA 12 "" 1 "" {XPPMATH 262 "6#,$*&,J-%' RootOfG6$,L*$%#_ZG\"#?\"\"\"*$F+\"#>\"\"#*$F+\"#=!#l*$F+\"#M^G>'=rn3;l??r3`u*GoJ*\"^q_! \\75)*HL)=K'>#zEL#e1A'yJ,LL7)3:-&4Jdh)z2.MK&*QQf3MAB-M[*$F&FV#!_qNb$[( Qbg0-8[R1(4AmF/BB+*=8'o&*Q*>wzWk$\\#GRu\"3e:Z%>GvR9b=\"]qU[(o,L))QJ?F. K6APwn.J')o@A(o9Dq\"\\=i-Jj%QB()eJ(*)4o0(Qqc!)*$F&Fin#\"^q%[%y#*=5W8\" f-#4V)y$RV*y(Hy#*[:J-lZu'*G$4r^Es&>Eyv>\\#))HUDO*)Fbo*$F&F/#\"[qIZ*\\4 \"[kW,,dx'QuK@M8KBxlyd33%=%3Na/dq/Bv*p$\\J()\\>$3P>Fbo#\"\\qzQ2#o]hEO] sO!yC6EgQFfWh%f#flN7\"f&fUHa\"G#)=%eScA;?$*=_,\"FfoF-*$F&FS#\"`q\"e:1J 8h`;i\"pC/'zA)H!f;;(QKSGQ8dtoNTqX\"^qEXi]!*\\mT4;)4 'Rj;\"H.6$*e1lmhSa2^Zb'yI**Q:qhw%>pH/ \\,D9tnBCKec@\"\\`s:Ypo9jhO&z\")4CenlQZ%\"^q%o\\P.mwxiSa1kAWu_N2isPVWu $H]S$)pV_?m#pnWxJYz>O6u2M6;*$F&F>#\"^qX&>zOJ'z0gaK5L2T0 U!z!y&[o\"*oN@H=VNAkvVA$Fbo*$F&FD#!_qT&RWrFlWOGVXS[toN[6km@n1&G)*38=(y DtC5#HA<\\b,sLH*3\"zimTFfp*$F&FG#\"`qRhD)\\H=\\a46\">&Gy=^*fn]r0,oO2yn )*e_8\"e(zw4$)\\gNTuiiL4PS*>Ffp*$F&FJ#\"`qn$*\\!*\\@>98>'Hp&*GL=-tLN@( *R%>Idx\"RkstTk4SnT(>6nOV)*oY=`,#Ffo*$F&FA#!^qJ#y!G[_PMNPb!G,#ybgdRQ&p jb(\\p!*[.5W#fsg,()*=*[JyQX#)emY(Q$\"]q4Y/vqg(HNaT'o&f(z?XO4U!zh(HVn>A [#=c]*\\Q:7!pnlS;$pVlVE<*$F&F5#!]qi`-X^RULB.t>@v7df&3$p%)yhOr6!y'[s!=W R!GM*=k.)*o[\\[,G!)G\"Fbo*$F&F2#\"[qt,k5uH&)zVz;if+g5m? #fQ/H5)e'QXGUc[XfR #[qst-uqw:HMW=UN(Ffo*$F&F;#\"^q4$e@sf%)=g@'*3rtZ^*4G7q`6z'39!R>**fD(*f $z>(zs*=PcXA?RRN#o#Fbo*$F&FY#!_q-3_sh\"eh[X%HtT&=E>1`/hPg+`o@!HAp!QF+B m7&\\!z0X9+GLfOFO$\"]q@uV3lT%p:gj,m0h=)Q=bJW36OMd7&eC4J^lJ#phVzl[\\SGN >NGS*$F&Ffn#\"_qw$*HGh)*yl!)=Z=$)e:>B=Y8%*[\\ZDj(pW0Q9()o^[9rks*GyFZYb K%Q:$FboF-)F&,&%\"nGF-F-F-!\"\"Fds" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "As observed in [Fin97], if " }{XPPEDIT 18 0 "g(n)" "-%\"gG6#%\" nG" }{TEXT -1 36 " denotes the number of tilings of a " }{XPPEDIT 18 0 "nxn" "I$nxnG6\"" }{TEXT -1 67 " chessboard, an interesting value fo r the physical applications is " }{XPPEDIT 18 0 "tau=Limit((g(n))^(1/n ^2),n=infinity)" "/%$tauG-%&LimitG6$)-%\"gG6#%\"nG*&\"\"\"\"\"\"*$F+\" \"#!\"\"/F+%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 219 "No exact expression for this limit is known, although the approxi mation 1.940215531 is generally agreed on. The first terms of the seq uence can be computed from the previous approximations and are consist ent with 1.94:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "nn:='nn': " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "for i from 1 to 6 do as sign(nn[i],coeff(series(gf[i],z=0,i+1),z,i)),evalf((nn[i])^(1/(i*i))) \+ od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"\"\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+ildE;!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ Xp!*=L=!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "But more \+ interesting is the following observation. Suppose for example " } {XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 31 " is a multiple of 6. To til e a " }{XPPEDIT 18 0 "nxn" "I$nxnG6\"" }{TEXT -1 36 " chessboard we ca n put side by side " }{XPPEDIT 18 0 "n/6" "*&%\"nG\"\"\"\"\"'!\"\"" } {TEXT -1 33 " slices of width 6. In this case " }{XPPEDIT 18 0 "tau=al pha^(1/6)" "/%$tauG)%&alphaG*&\"\"\"\"\"\"\"\"'!\"\"" }{TEXT -1 7 " w ith " }{XPPEDIT 18 0 "alpha" "I&alphaG6\"" }{TEXT -1 59 " the singular ity of smallest modulus of the denominator of " }{XPPEDIT 18 0 "gf[6] " "&%#gfG6#\"\"'" }{TEXT -1 5 ". If " }{XPPEDIT 18 0 "n" "I\"nG6\"" } {TEXT -1 281 " is not a multiple of 6, it suffices to complete with at most 5 vertical stripes of width 1, but this does not change the limi t. The interest in using as many slices of maximal width is to minimi ze the number of joints where the overlaps are not taken into account. The sequence " }{XPPEDIT 18 0 "\{alpha[i]^(1/i),i=1..6\}" "<$)&%&alp haG6#%\"iG*&\"\"\"\"\"\"F'!\"\"/F';\"\"\"\"\"'" }{TEXT -1 50 " therefo re provides lower bounds for the constant " }{XPPEDIT 18 0 "tau" "I$ta uG6\"" }{TEXT -1 134 ". An upper bound can be obtained in the same way by having slices of width 6 overlap on a position, and the correspond ing sequence is " }{XPPEDIT 18 0 "\{alpha[i]^(1/(i-1)),i=2..6" "<$)&%& alphaG6#%\"iG*&\"\"\"\"\"\",&F'F*\"\"\"!\"\"F-/F';\"\"#\"\"'" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "for i from 2 t o 6 do i,(1/op(1,denom(evalf(asGf[i]))))^(1./i),(1/op(1,denom(evalf(as Gf[i]))))^(1./(i-1)) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"#$\"+/ C&Gz\"!\"*$\"+V(>V@$F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"$$\"+N>G Q=!\"*$\"+0DS#\\#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"%$\"+5q\\j= !\"*$\"+V1=$H#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"&$\"+FRcy=!\"* $\"+m)*G*>#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"'$\"+()\\q))=!\"* $\"+N,&[9#F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "At last a trick w e can use to try to guess the value of " }{XPPEDIT 18 0 "tau" "I$tauG6 \"" }{TEXT -1 44 " is Romberg's convergence acceleration. Let " } {XPPEDIT 18 0 "u[n]" "&%\"uG6#%\"nG" }{TEXT -1 36 " be a sequence know n to converge to " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 44 ". If th e rate of convergence is of the form " }{XPPEDIT 18 0 "u[n]=l+a[1]/n+O (1/n^2)" "/&%\"uG6#%\"nG,(%\"lG\"\"\"*&&%\"aG6#\"\"\"F)F&!\"\"F)-%\"OG 6#*&\"\"\"F)*$F&\"\"#F/F)" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "2*u[2 *n]-u[n]" ",&*&\"\"#\"\"\"&%\"uG6#*&\"\"#F%%\"nGF%F%F%&F'6#F+!\"\"" } {TEXT -1 4 " is " }{XPPEDIT 18 0 "l+O(1/n^2)" ",&%\"lG\"\"\"-%\"OG6#*& \"\"\"F$*$%\"nG\"\"#!\"\"F$" }{TEXT -1 178 ". On our example, although the upper bound does not make sense due to too erroneous initial valu es, after a single step the lower bound gets close to the commonly acc epted value:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "u[2]:=1.79 2852404:u[4]:=1.863497010:\nv[2]:=3.214319743:v[4]:=2.293180643:\n2*u[ 4]-u[2],2*v[4]-v[2];\n\n\nu[3]:=1.838281935:u[6]:=1.888704987:\nv[3]:= 2.492402505:v[6]:=2.144850135:\n2*u[6]-u[3],2*v[6]-v[3];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$$\"+;;9M>!\"*$\"+V:/s8F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+R!G\"R>!\"*$\"+lxH(z\"F%" }}}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 28 "Generating functions archive" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "gf[1]:=-1/(-1+z^2+z);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#gfG6#\"\"\",$*$,(!\"\"F'*$%\"zG\"\"#F'F-F'F+F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "gf[2]:=-1/(-3*z+1-z^2+z^3)*( z-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#gfG6#\"\"#,$*&,*%\"zG!\"$ \"\"\"F-*$F+F'!\"\"*$F+\"\"$F-F/,&F+F-F/F-F-F/" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 54 "gf[3]:=-(z^4+z^3-4*z^2-z+1)/(14*z^2-1+4*z+z^6- 10*z^4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#gfG6#\"\"$,$*&,,*$%\"z G\"\"%\"\"\"*$F,F'F.*$F,\"\"#!\"%F,!\"\"F.F.F.,,F0\"#9F3F.F,F-*$F,\"\" 'F.F+!#5F3F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "gf[4]:=-(1 -4*z-15*z^2+20*z^3+z^7-11*z^5-2*z^6+10*z^4)/(z^9-z^8-23*z^7+29*z^6+91* z^5-111*z^4-41*z^3+41*z^2+9*z-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> &%#gfG6#\"\"%,$*&,2\"\"\"F+%\"zG!\"%*$F,\"\"#!#:*$F,\"\"$\"#?*$F,\"\"( F+*$F,\"\"&!#6*$F,\"\"'!\"#*$F,F'\"#5F+,6*$F,\"\"*F+*$F,\"\")!\"\"F4!# BF9\"#HF6\"#\"*F " 0 "" {MPLTEXT 1 0 347 "gf[5]:=-(z^18+2*z^17-45*z^16-68*z^15+654*z^14+8 70*z^13-3820*z^12-4700*z^11+9255*z^10+9448*z^9-11175*z^8-7532*z^7+6956 *z^6+1994*z^5-1794*z^4-88*z^3+113*z^2+6*z-1)/(z^20+2*z^19-65*z^18-140* z^17+1281*z^16+2538*z^15-10366*z^14-17604*z^13+38553*z^12+50158*z^11-7 3623*z^10-60482*z^9+74665*z^8+26564*z^7-35106*z^6-898*z^5+4757*z^4+16* z^3-229*z^2-14*z+1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%#gfG6#\"\"& ,$*&,H*$%\"zG\"#=\"\"\"*$F,\"#<\"\"#*$F,\"#;!#X*$F,\"#:!#o*$F,\"#9\"$a '*$F,\"#8\"$q)*$F,\"#7!%?Q*$F,\"#6!%+Z*$F,\"#5\"%b#**$F,\"\"*\"%[%**$F ,\"\")!&v6\"*$F,\"\"(!%Kv*$F,\"\"'\"%cp*$F,F'\"%%*>*$F,\"\"%!%%z\"*$F, \"\"$!#))*$F,F1\"$8\"F,FQ!\"\"F.F.,L*$F,\"#?F.*$F,\"#>F1F+!#lF/!$S\"F2 \"%\"G\"F5\"%QDF8!&m.\"F;!&/w\"F>\"&`&QFA\"&e,&FD!&BO(FG!&#[gFJ\"&lY(F M\"&kl#FP!&1^$FS!$)*)FU\"%dZFXF3Fen!$H#F,!#9F.F.FgnFgn" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 882 "gf[6]:=-(-1+311*z^2-3891*z^3-12057 *z^4-315889*z^6-2997721*z^7+218447*z^5+13467571*z^9+8754480*z^8+23*z-4 58919487*z^18-303976032*z^17+612805499*z^16+207743591*z^15-496137395*z ^14-56233657*z^13+240612231*z^12-14684235*z^11-66016499*z^10+206819317 *z^20+249194245*z^19-109*z^32-36273*z^29+861*z^31+7443809*z^24+3722360 1*z^23-123372421*z^21-54160427*z^22-6708699*z^25+z^34-29377*z^28+68651 7*z^27-338040*z^26+3521*z^30-7*z^33)/(1-576*z^2+6080*z^3+42422*z^4-443 404*z^6+12931566*z^7-453004*z^5-83558644*z^9-25517604*z^8-36*z+4169343 006*z^18+2978277152*z^17-4669345206*z^16-1630080704*z^15+3235975264*z^ 14+274712602*z^13-1335612340*z^12+154307596*z^11+295510396*z^10-231032 7672*z^20-2919950172*z^19+5736*z^32+1503868*z^29-62874*z^31-149620588* z^24-626694028*z^23+1717916424*z^21+777289050*z^22+141424642*z^25-8*z^ 35-138*z^34+z^36-94620*z^28-19237868*z^27+13835164*z^26-81796*z^30+122 4*z^33);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%#gfG6#\"\"',$*&,bo*$%\" zG\"#=!*([>*e%*$F,\"#:\"*\"fVx?*$F,\"#o?*$F,\"#>\"*XU>\\#*$F, \"#5!)*\\;g'*$F,\"#9!*&RPh\\*$F,\"#8!)dOBc*$F,\"#7\"*JAhS#*$F,\"#6!)NU o9*$F,\"#;\"**\\0Gh*$F,\"#K!$4\"*$F,\"#H!&ti$*$F,\"#J\"$h)*$F,\"#C\"(4 QW(*$F,F5\"),OAP*$F,\"#A!)F/;a*$F,\"#D!(*p3n*$F,\"#M\"\"\"*$F,\"#G!&x$ H*$F,\"#F\"'HFS\"*'R5b HFV\"+k_(fB$FY\"*-Eru#Ffn!+SBhN8Fin\"*'f2V:F\\o!+1_MpY*$F,\"#N!\")*$F, \"#OFepF_o\"%OdFbo\"(oQ]\"Feo!&uG'Fho!*)e?'\\\"F[p!*GSpE'F]p\"*]!*Gx(F `p\"*UYUT\"Fcp!$Q\"Ffp!&?Y*Fip!)oyB>F\\q\")k^$Q\"F_q!&'z\")Fbq\"%C7Feq \"+Ck\"zr\"FepFepFhqFhq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Co nclusion" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 191 "We showed that variou s parameters related to dimer-monomer tilings such as the average numb er of pieces or the relative numbers of horizontal dimers and monomers in a random tiling of height " }{XPPEDIT 18 0 "n " "I\"nG6\"" }{TEXT -1 21 " in a strip of width " }{XPPEDIT 18 0 "m" "I\"mG6\"" }{TEXT -1 251 " can be computed very easily using Combstruct and ratasympt. More precisely Combstruct is used to define the grammars the tilings are d erived from, and ratasympt is used to perform asymptotic expansions on rational fractions with rational coeficients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "About the number " } {XPPEDIT 18 0 "g(n)" "-%\"gG6#%\"nG" }{TEXT -1 27 " of different tilin gs of a " }{XPPEDIT 18 0 "nxn" "I$nxnG6\"" }{TEXT -1 167 " chessboard, altough the method presented here is limited due to the exponential g rowth of the grammar describing these tilings, the very first terms co mputed provide " }{TEXT 264 36 "provably good upper and lower bounds" }{TEXT -1 31 " for the connectivity constant " }{XPPEDIT 18 0 "g(n)^(1 /n^2)" ")-%\"gG6#%\"nG*&\"\"\"\"\"\"*$F&\"\"#!\"\"" }{TEXT -1 17 ". Mo re precisely:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 7 "Theorem" }{TEXT -1 80 ". The connectvity constant for two dimensional monomer-dimer tilings satisfies " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "tau>=1.888" "1$\"%)) =!\"$%$tauG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "tau<=2.144" "1%$tauG$ \"%W@!\"$" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "3 5 0 0" 50 }{VIEWOPTS 1 1 0 1 1 1803 } PARA 0 "" 0 "" {TEXT -1 17 "About the number " } {XPPEDIT 18 0 "g(n)" "-%\"gG6#%\"nG" }{TEXT -1 27 " of different tilin gs of a " }{XPPEDIT 18 0 "nxn" "I$nxnG6\"" }{TEXT -1 167 " chessboard, altough the method presented here is limited due to the exponential g rowth of the grammar describing these tilings, the very first terms co mputed provide " }{TEXT 264 36 "provably good upper and lower bounds" }{TEXT -1 31 " for the connectivity constant " }{XPPEDIT 18 0 "g(n)^(1 /n^2)" ")-%\"gG6#%\"nG*&\"\"\"\"\"\"*$F&\"\"#!\"\"" }{TEXT -1 17 ". Mo re precisely:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 7 "Theorem" }{TEXT -1 80 ". The connectvity constant for two dim+wenumeratalcoholotherclasschemicalmoleculexamplpolyatheorfredericchyzakversjanuaralkansimplclascompoundtheygenericaldescribformulafirstsmallmethanethanpropanbutangivenhowevexistseveraldifferisomerstructurbondbetweenatomchemistrmuchinterestknownumbbettyetlistsuchobtainreplachydrogengroupfollowisomorphiccarbonchainwithdistinguishnodeagainalkylradicalmisswedisregardgeometricalconstraintconsidstructuralonlyconformationalleadpuregraphtheoreticalproblemhowmanyroottreeinternaleachdegresessthuscountgenericcombinatoricalsocorrespondorganometalicanymonosubstitutnexttreatcasedisubstituttrisubstitutdevelopstudourmodelusingpackagcombstructpartreferbookreadcombinatorialspringverlagmoreextensresultsectwithoutaccordheightgeneraldefinitcanviewlinkatmosttakeintoaccountimplicitnolossinformatsincalwayrecoverskeletonyieldequatmapgrammargrammprodsetcardspecunlabellnoteconstsnumgfundiffeqtoproccreatmaplprocedurdifferentialequatcallsequenceqprecprecisdisklistparameterlinearwithpolynomialcoefficientnamefunctvariabloptionalpositintegnumbdigitabsolutformpathradiudescriptcommandreturnsuchmaypointevaluatatmorepreciseequivalevaldiffeqoptiongivenperformprecomputatmakesubsequupwithinfastelementstartoriginavoidsingularwhosendpointgivecentprecomputdatausedpossiblsinglopposonlycasethuschoicdeterminatmultivalufunctiondonetimeexamplgfunholexprtodiffeqairyaiplotinfolevelsteptransitmatrixderivatwarnrecalworkerroralsoending 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-1 -'1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Error" 7 8 1 {CSTYLE "" -1 -1 "" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Dash Item" 0 16 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 16 3 } {PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 5 257 1 {CSTYLE "" -1 -1 " " 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -(1 -1 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 256 21 "BALLS AND URNS, ETC.\n" } {TEXT 268 1 "\n" }{TEXT 257 17 "Philippe Flajolet" }}{PARA 262 "" 0 " " {TEXT -1 30 "(Version of December 14, 1996)" }}{PARA 263 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 274 "Balls and urns models ar e basic in combinatorics, statistics, analysis of algorithms, and stat istical physics. These models are nicely decomposable and their basic \+ properties can be explored using tools developed for the automatic man ipulation of combinatorial models, like " }{HYPERLNK 17 "Combstruct" 2 "combstruct" "" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 125 "As i )s well-known there are four types of models, depending on whether ball s and urns are taken to be distinguishable or not." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "The four basic models" }}{PARA 0 "" 0 "" {TEXT -1 350 "We consider the placement of balls into urns in all possible ways . For definiteness, we examine only the situation of nonempty urns, so that the number of possible configurations of a fixed size (i.e., a f ixed number of balls) is always finite. If the balls are distinguishab le, we may assume them to be numbered consecutively by integers 1, 2, \+ ..., " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 200 "; in this case, we are dealing with labelled structures, and balls are labelled atoms. I f the balls are indistinguishable, then we simply regard them as anony mous unlabelled atoms (generically called " }{HYPERLNK 17 "Z" 2 "combs truct[specification]" "" }{TEXT -1 143 ", by a global convention of Co mbstruct). If the urns are d*istinguishable, we may view them as arrang ed in a row, so that we are dealing with a " }{HYPERLNK 17 "Sequence" 2 "combstruct[specification]" "" }{TEXT -1 36 " construction; otherwis e, we have a " }{HYPERLNK 17 "Set" 2 "combstruct[specification]" "" } {TEXT -1 20 " construction. (The " }{HYPERLNK 17 "Set" 2 "combstruct[s pecification]" "" }{TEXT -1 98 " construction of Combstruct means a mu ltiset, that is to say a set where repetitions are allowed.)" }}{PARA 0 "" 0 "" {TEXT -1 66 "Balls are not ordered within an urn, so that an urn is a priori a " }{HYPERLNK 17 "Set" 2 "combstruct[specification] " "" }{TEXT -1 52 " of balls. This gives rise to four different models :" }}{PARA 16 "" 0 "" {TEXT -1 116 "DBDU: distinguishable balls and di stinguishable urns; we are dealing with Sequences of Sets, in a labell ed universe;" }}{PARA 16 "" 0 "" {TEXT -1 113 "DBIU: distinguishable b alls and indistinguishable urns; we are dealing with Sets of Sets, in \+ a labelled universe;" }}{PARA 16 "" 0 "" {TEXT -1 121 +"IBDU: indisting uishable balls and distinguishable urns; we are dealing with Sequences of Sets, in an unlabelled universe;" }}{PARA 16 "" 0 "" {TEXT -1 118 "IBIU: indistinguishable balls and indistinguishable urns; we are deal ing with Sets of Sets, in an unlabelled universe." }}{PARA 0 "" 0 "" {TEXT -1 88 "In combstruct, this is expressed by four different, but s imilar looking, specifications:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "with(combstruct);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.%+allst ructsG%&countG%%drawG%)finishedG%'gfeqnsG%)gfseriesG%(gfsolveG%,iterst ructsG%+nextstructG%,prog_gfeqnsG%.prog_gfseriesG%-prog_gfsolveG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "DBDU:=[S,\{S=Sequence(U),U=S et(Z,card>=1)\},labelled]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "DBIU:=[S,\{S=Set(U),U=Set(Z,card>=1)\},labelled]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "IBDU:=[S,\{S=Sequence(U),U=Set(Z,ca rd>=1)\},unlabelled]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "IB IU:=[S,\{S=Set(U),,U=Set(Z,card>=1)\},unlabelled]:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 57 "for spec in DBDU,DBIU,IBDU,IBIU do draw(spec ,size=10) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%)SequenceG6*-%$SetG 6#&%\"ZG6#\"\"%-F'6#&F*6#\"\"$-F'6#&F*6#\"\"&-F'6$&F*6#\"\"*&F*6#\"\"( -F'6#&F*6#\"\"#-F'6#&F*6#\"\")-F'6#&F*6#\"\"\"-F'6$&F*6#\"#5&F*6#\"\"' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SetG6&-F$6#&%\"ZG6#\"\"&-F$6%&F )6#\"\"#&F)6#\"\")&F)6#\"#5-F$6%&F)6#\"\"$&F)6#\"\"*&F)6#\"\"(-F$6%&F) 6#\"\"%&F)6#\"\"\"&F)6#\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%)Seq uenceG6%-%$SetG6$%\"ZGF)-F'6)F)F)F)F)F)F)F)-F'6#F)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$SetG6'-F$6#%\"ZGF&-F$6$F(F(F)-F$6&F(F(F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 161 "The corresponding counting sequences sat isfy natural domination conditions that one can summarize by the infor mal inequality: \"Distinguishable>Indistinguishable\"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "for spec in DBDU,DBIU,IBDU,IBIU do seq(co unt(spec,size=j),j=1..12) od;" }}{PARA 11 "" 1 "-" {XPPMATH 20 "6.\"\" \"\"\"$\"#8\"#v\"$T&\"%$o%\"&$HZ\"'Nea\"(hs3(\"*jvC-\"\"+tDjA;\",&fn:4 G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6.\"\"\"\"\"#\"\"&\"#:\"#_\"$.#\"$x )\"%ST\"&Z6#\"'vf6\"'q&y'\"((f8U" }}{PARA 11 "" 1 "" {XPPMATH 20 "6.\" \"\"\"\"#\"\"%\"\")\"#;\"#K\"#k\"$G\"\"$c#\"$7&\"%C5\"%[?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6.\"\"\"\"\"#\"\"$\"\"&\"\"(\"#6\"#:\"#A\"#I\"#U \"#c\"#x" }}}{PARA 0 "" 0 "" {TEXT -1 158 "In the sequel, it is conven ient to represent objects by a more concise notation. We thus introduc e \"reduction\" procedures for labelled and unlabelled objects:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 177 "lreduce:=proc(e) eval(subs( \{Set=proc() \{args\} end, Sequence=proc() [args] end\},e)) end:\nured uce:=proc(e) eval(subs(\{Set=proc() \{[args]\} end, Sequence=proc() [a rgs] end\},e)) end:" }}}{PARA 0 "" 0 "" {TEXT -1 124 "Since the set co nstruction \"\{\}\" in Maple does not keep multisets, an unlabelled (m ulti)set will be represented as \"\{[...]\}\"." }}{EXCHG {PARA 0 "> " 0 "" {MP.LTEXT 1 0 113 "for spec in DBDU,DBIU do lreduce(draw(spec,size =25)) od;\nfor spec in IBDU,IBIU do ureduce(draw(spec,size=25)) od;" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#79<#&%\"ZG6#\"#C<#&F&6#\"#6<#&F&6#\"# 9<#&F&6#\"\"&<#&F&6#\"#D<$&F&6#\"\"(&F&6#\"#?<#&F&6#\"\"#<#&F&6#\"\"%< #&F&6#\"#8<#&F&6#\"#<<#&F&6#\"\"\"<#&F&6#\"\"$<#&F&6#\"#=<#&F&6#\"#5<# &F&6#\"#B<#&F&6#\"\"*<#&F&6#\"#@<#&F&6#\"#><#&F&6#\"#A<$&F&6#\"\")&F&6 #\"#:<#&F&6#\"#7<#&F&6#\"\"'<#&F&6#\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<-<#&%\"ZG6#\"\"#<#&F&6#\"#><$&F&6#\"\"*&F&6#\"#@<$&F&6#\"\"\"&F &6#\"#6<$&F&6#\"\"%&F&6#\"#?<$&F&6#\"#C&F&6#\"#7<&&F&6#\"\"$&F&6#\"#5& F&6#\"#A&F&6#\"#:<$&F&6#\"#8&F&6#\"#B<%&F&6#\"\")&F&6#\"#<&F&6#\"#=<&& F&6#\"\"&&F&6#\"\"(&F&6#\"#9&F&6#\"#;<$&F&6#\"\"'&F&6#\"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-<#7(%\"ZGF&F&F&F&F&<#7#F&F'F'<#7%F&F&F&F)< #7$F&F&F'F'F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#7&<#7%%\"ZGF'F'F% <#73F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'<#7$F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 105 "On such simulations, we se/e that there tends to be fewer urns in models of type IU, but more filled ones." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 258 43 "Distinguishable balls (labelled structures)" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Distinguishable urns" }}{PARA 0 " " 0 "" {TEXT -1 156 "In this model, we deal with distinguishable balls (labelled atoms) that go in all possible way into distinguishable urn s corresponding to the specification:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "DBDU:=[S,\{S=Sequence(U),U=Set(Z,card>=1)\},labelled] :" }}}{PARA 0 "" 0 "" {TEXT -1 63 "Combinatorially, this model is the \+ same as of Surjections from " }{XPPEDIT 18 0 "[1..n]" "7#;\"\"\"%\"nG " }{TEXT -1 94 " to an initial segment of the integers. It is the one \+ that leads to larger cardinality counts." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "for j to 3 do j=map(lreduce,allstructs(DBDU,size=j)) \+ od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"\"7#7#<#&%\"ZG6#F$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"#7%7$<#&%\"ZG6#F$<#&F)6#\"\"\"7$F 0+F'7#<$F(F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/\"\"$7/7#<%&%\"ZG6#F$& F)6#\"\"#&F)6#\"\"\"7%<#F.<#F(<#F+7%F4F2F37$<$F(F+F27$F3<$F+F.7$<$F(F. F47$F2F77$F4F;7%F3F2F47%F3F4F27%F4F3F27$F9F37%F2F4F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(count(DBDU,size=j),j=0..30);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6A\"\"\"F#\"\"$\"#8\"#v\"$T&\"%$o%\"&$HZ \"'Nea\"(hs3(\"*jvC-\"\"+tDjA;\",&fn:4G\"-\"Q[$eo_\"/V/(HMT1\"\"0`y(4> $GI#\"1b8)>oacJ&\"3,f8Hqwq.8\"4B`%oDjY`&Q$\"5L6TG$>te,G*\"7:J?%QWiz(ox E\"8@=)Q2/&)*\\#[7\")\":.Ua%Q!>.)>W%[d#\";8/e%H4duO8XQa)\"=v)[1lsa9u57 z#eH\"?T\")\\$eDexv%QaO(p1\"\"@$3Rv7'[#\\oT%)fdA-S\"B$p`iAw#\\Ki@'=Rw( *e:\"CNYLNRP=N.m+&\\1ivH'\"EhodW!HL*3-!pBIE&QyME\"Gj>!\\='>kCu$[!)=,%z oNS6" }}}{PARA 0 "" 0 "" {TEXT -1 133 "Such tables are quite useful fo r checking various combinatorial conjectures. Here, we may verify that these numbers are the sequence " }{TEXT 269 5 "M2952" }{TEXT -1 8 " o f the " }{TEXT 270 33 "Encyclopedia of Integer Sequences" }{TEXT -1 92 " by Sloane and Plouffe, where1 they are known as the numbers of pre ferential arrangements of " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 57 " things.\nThe counting problem is solved automatically by " } {HYPERLNK 17 "combstruct[gfeqns]" 2 "combstruct[gfeqns]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "combstruct[gfseries]" 2 "combstruct[gfseries]" " " }{TEXT -1 26 " (a series alternative to " }{HYPERLNK 17 "combstruct[ count]" 2 "combstruct[count]" "" }{TEXT -1 6 ") and " }{HYPERLNK 17 "c ombstruct[gfsolve]" 2 "combstruct[gfsolve]" "" }{TEXT -1 1 ":" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "gfeqns(op(2..3,DBDU),z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%/-%\"UG6#%\"zG,&-%$expG6#-%\"ZGF'\" \"\"!\"\"F//-%\"SGF'*$,&F/F/F%F0F0/F-F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Order:=12: gfseries(op(2..3,DBDU),z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7%/-%\"SG6#%\"zG+=F+\"\"\"\"\"!F-\"\"\" #\"\"$\"\"#\"\"##\"#8\"\"'\"\"$#\"#D\"\")\"\"%#\"$T&\"$?\"\"\"&#\"%h: \"$S#\"\"'#\"&$HZ\"%S]\"\"(#\"&*QO\"%)o#\"\")#\"(hs3(\"'!)GO\"\"*#\")@ D3M2\"(+'47\"#5#\"+tDjA;\")+o\"*R\"#6-%\"OG6#F-\"#7/-%\"UGF*+;F+F-\"\" \"#F-F2\"\"##F-F6\"\"$#F-\"#C\"\"%#F-F>\"\"&#F-\"$?(\"\"'#F-FF\"\"(#F- \"&?.%\"\")#F-FN\"\"*#F-\"(+)GO\"#5#F-FV\"#6FX\"#7/-%\"ZGF*+%F+F-\"\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "gfsolve(op(2..3,DBDU) ,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%\"SG6#%\"zG,$*$,&!\"#\"\" \"-%$expGF'F-!\"\"F0/-%\"ZGF'F(/-%\"UGF',&F.F-F0F-" }}}{PARA 0 "" 0 " " {TEXT -1 82 "In particular, we have found the exponential generating function (EGF) explicitly:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "S_z:=subs(\",S(z)); series(S_z,z=0,7);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$S_zG,$*$,&!\"#\"\"\"-%$ex pG6#%\"zGF)!\"\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"zG\"\"\"\" \"!F%\"\"\"#\"\"$\"\"#\"\"##\"#8\"\"'\"\"$#\"#D\"\")\"\"%#\"$T&\"$?\" \"\"&#\"%h:\"$S#\"\"'-%\"OG6#F%\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 35 " The EGF is singular with a pole at " }{XPPEDIT 18 0 "z=ln(2)" "/%\"zG- %#lnG6#\"\"#" }{TEXT -1 72 ". This im3mediately gives an approximate ex pression for the coefficients:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "series(S_z,z=log(2),3);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+),&%\"zG\"\"\"-%#lnG6#\"\"#!\"\"#F+F* !\"\"#F&\"\"%\"\"!-%\"OG6#F&\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "S_n_asympt:=1/2*n!*log(2)^(-n-1);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%+S_n_asymptG,$*&-%*factorialG6#%\"nG\"\"\")-%#lnG6# \"\"#,&F*!\"\"F2F+F+#F+F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "As u sual with meromorphic functions, the approximation is extremely good: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "for j from 0 by 5 to 30 do j,ev alf(count(DBDU,size=j)/subs(n=j,S_n_asympt),30); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!$\"?#HCkW$)=1*)>6O%H'Q\"!#H" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$\"\"&$\"?W)ej0U#=TLQJ>(*****!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#5$\"?M>*)[b@Q#[%[**********!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#:$\"?y\"*QtZVP)***************!#I" }}{PARA 11 "" 1 " " {XPPMA4TH 20 "6$\"#?$\"?X;%f-,+++++++++\"!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#D$\"?p5-+++++++++++5!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#I$\"?\"*****************************!#I" }}}{PARA 0 "" 0 "" {TEXT -1 259 "This type of analysis can be easily generalized \+ to determine for instance the expected number of urns in a random surj ection. Such analyses may then be used to validate an a priori statist ical model by comparing theoretical predictions against empirical data ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "Indistinguishable urns (set partitions)" }}{PARA 0 "" 0 "" {TEXT -1 82 "We are now dealing with indistinguishable urns. Equivalen tly, we consider the way " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 30 " elements (the labels 1, ..., " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 63 ") may be grouped into equivalence classes in all possible ways. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "DBIU:=[S,\{S=Set(U),U=Se t(Z,card>=1)\},labelled]:" }}}{EXCHG {PARA 0 "> " 50 "" {MPLTEXT 1 0 56 "for j to 4 do j=map(lreduce,allstructs(DBIU,size=j)) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"\"7#<#<#&%\"ZG6#F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"#7$<$<#&%\"ZG6#\"\"\"<#&F)6#F$<#<$F-F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"$7'<$<#&%\"ZG6#\"\"\"<$&F)6#F$&F)6#\"\" #<$<$F/F(<#F-<#<%F-F/F(<$<#F/<$F-F(<%F'F8F4" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/\"\"%71<$<%&%\"ZG6#\"\"$&F)6#\"\"#&F)6#\"\"\"<#&F)6#F$ <%<#F,<#F(<$F/F3<%<#F/F6<$F(F3<$F6<%F(F/F3<$F:<%F(F,F3<&F:F6F7F2<%<$F, F/F7F2<%F:F7<$F,F3<#<&F(F,F/F3<$FBF;<$F7<%F,F/F3<$<$F(F/FD<%F:<$F(F,F2 <$FMF8<%F6FKF2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(count (DBIU,size=j),j=0..30);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6A\"\"\"F#\" \"#\"\"&\"#:\"#_\"$.#\"$x)\"%ST\"&Z6#\"'vf6\"'q&y'\"((f8U\")PWkF\"*A$* *3>\"+X&eHQ\"\",Z@9![5\",/)p['G)\"-fh!o2#o\".d]?UF$e\"/s`BeTs^\"0^n:;) p[Z\"1BtWQdr1X\"2YV3be+_T%\"3*G0[Hp)efW\"4`$***HAL!fQY\"5uiv=O_Y7j\\\" 6*Q*)*fg$z/v-')Q>!)R8(\"9Z,XK$4=^9!\\n%)" }}} {PARA 0 "" 0 "" 6{TEXT -1 133 "Such tables are quite useful for checkin g various combinatorial conjectures. Here, we may verify that these nu mbers are the sequence " }{TEXT 271 5 "M1484" }{TEXT -1 8 " of the " } {TEXT 259 33 "Encyclopedia of Integer Sequences" }{TEXT -1 48 " by Slo ane and Plouffe. They are the well-known " }{HYPERLNK 17 "Bell numbers " 2 "combinat[bell]" "" }{TEXT -1 125 " of combinatorial theory that a lso appear as moments of the Poisson distribution, in the calculus of \+ finite differences, etc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 62 "We automatically obtain the exponential generating function as" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "gfsolve(op(2 ..3,DBIU),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%\"ZG6#%\"zGF(/-% \"UGF',&-%$expGF'\"\"\"!\"\"F//-%\"SGF'-F.6#F," }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "P_z:=subs(\",S(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$P_zG-%$expG6#,&-F&6#%\"zG\"\"\"!\"\"F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "series(P_z,z=70,8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+5%\"zG\"\"\"\"\"!F%\"\"\"F%\"\"##\"\"&\"\"'\"\"$# F*\"\")\"\"%#\"#8\"#I\"\"&#\"$.#\"$?(\"\"'#\"$x)\"%S]\"\"(-%\"OG6#F%\" \")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Order:=8: gfseries(o p(2..3,DBIU),z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7%/-%\" SG6#%\"zG+5F+\"\"\"\"\"!F-\"\"\"F-\"\"##\"\"&\"\"'\"\"$#F2\"\")\"\"%# \"#8\"#I\"\"&#\"$.#\"$?(\"\"'#\"$x)\"%S]\"\"(-%\"OG6#F-\"\")/-%\"UGF*+ 3F+F-\"\"\"#F-\"\"#\"\"##F-F3\"\"$#F-\"#C\"\"%#F-\"$?\"\"\"&#F-F>\"\"' #F-FB\"\"(FD\"\")/-%\"ZGF*+%F+F-\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 131 "By expanding and truncating, we obtain excellent approximations ( this is in fact a version of a formula found by Dobinski in 1877):" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "P_n_asympt:=exp(-1)*Sum(k^n /k!,k=0..2*n);\nfor j by 3 to 20 do j,evalf(count(DBIU,size=j)/subs(n= j,P_n_asympt),30); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+P_n_asymp tG*&-%$expG6#!\"\"\"\"\"-%$SumG6$*&)%\"kG%\"nGF*-%*factorialG6#F0F)/F0 ;\"\"!,8$F1\"\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"$\"?oNP9!o< E_HU\"49f8!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"%$\"?=z)pS%o]A'[Y Y@0+\"!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"($\"?d8Vx^jn#H<21+++ \"!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#5$\"?fRk7d&yf6,+++++\"!#H " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#8$\"?'GC0GD0++++++++\"!#H" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"#;$\"?C$R3+++++++++++\"!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#>$\"?,+++++++++++++5!#H" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 47 "Indistinguishable balls (unlabelled struc tures)" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "Distinguishable urns (i nteger compositions)" }}{PARA 0 "" 0 "" {TEXT -1 31 "We start from the specification" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "IBDU:=[S, \{S=Sequence(U),U=Sequence(Z,card>=1)\},unlabelled]:" }}}{PARA 0 "" 0 "" {TEXT -1 89 "In this particular case, as balls are indistinguishabl e, we may as well consider urns as " }{HYPERLNK 17 "Sequence" 2 "combs truct[specification]" "" }{TEXT -1 250 " 9of atoms. The reason for doin g this is a simpler form of generating functions (as we do not have to go unnecessarily through Polya operators) as well as faster computati ons. We can check that this new version is equivalent to the earlier o ne, namely" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "IBDU_0:=[S,\{S =Sequence(U),U=Set(Z,card>=1)\},unlabelled]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "seq(count(IBDU,size=j),j=0..20); seq(count(IBDU_ 0,size=j),j=0..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "67\"\"\"F#\"\"#\" \"%\"\")\"#;\"#K\"#k\"$G\"\"$c#\"$7&\"%C5\"%[?\"%'4%\"%#>)\"&%Q;\"&oF$ \"&Ob'\"'s58\"'W@E\"')GC&" }}{PARA 11 "" 1 "" {XPPMATH 20 "67\"\"\"F# \"\"#\"\"%\"\")\"#;\"#K\"#k\"$G\"\"$c#\"$7&\"%C5\"%[?\"%'4%\"%#>)\"&%Q ;\"&oF$\"&Ob'\"'s58\"'W@E\"')GC&" }}}{PARA 0 "" 0 "" {TEXT -1 91 "Of c ourse, here we recognize the powers of two: the result is combinatoria lly obvious since" }}{PARA 0 "" 0 "" {TEXT -1 88 "a partition can be o btained by inserting arbitrary cuts in the integer interval 1, ..., ": }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 49 ". We can also check this w ith combstruct[gfsolve]" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "g fsolve(op(2..3,IBDU),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%\"ZG6 #%\"zGF(/-%\"SGF'*&,&!\"\"\"\"\"F(F/F/,&F.F/F(\"\"#F./-%\"UGF',$*&F(F/ F-F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "SS_z:=subs(\",S(z ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%SS_zG*&,&!\"\"\"\"\"%\"zGF(F (,&F'F(F)\"\"#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "series( SS_z,z=0,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+9%\"zG\"\"\"\"\"!F% \"\"\"\"\"#\"\"#\"\"%\"\"$\"\")\"\"%\"#;\"\"&\"#K\"\"'\"#k\"\"(\"$G\" \"\")\"$c#\"\"*-%\"OG6#F%\"#5" }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT 260 43 "Indistinguishable urns (integer partitions)" }}{PARA 0 "" 0 " " {TEXT -1 31 "We start from the specification" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "IBIU:=[S,\{S=Set(U),U=Sequence(Z,card>=1)\},unla belled]:" }}}{PARA 0 "" 0 "" {TEXT -1 99 "Combinatorially, we are spec ifying integer partitions tha;t describe the occupancy profile of urns ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "for j to 6 do j=map(ure duce,allstructs(IBIU,size=j)) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ \"\"\"7#<#7#7#%\"ZG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"#7$<#7$7#% \"ZGF(<#7#7$F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"$7%<#7#7%%\"Z GF)F)<#7%7#F)F,F,<#7$7$F)F)F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\" %7'<#7$7$%\"ZGF)F(<#7$7%F)F)F)7#F)<#7%F(F-F-<#7&F-F-F-F-<#7#7&F)F)F)F) " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/\"\"&7)<#7$7%%\"ZGF)F)7$F)F)<#7%F *F*7#F)<#7&F*F-F-F-<#7#7'F)F)F)F)F)<#7$F-7&F)F)F)F)<#7%F(F-F-<#7'F-F-F -F-F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/\"\"'7-<#7(7#%\"ZGF(F(F(F(F( <#7$7%F)F)F)F,<#7&F,F(F(F(<#7&7$F)F)F1F(F(<#7%F1F1F1<#7%F(F(7&F)F)F)F) <#7'F1F(F(F(F(<#7%F,F1F(<#7$F(7'F)F)F)F)F)<#7#7(F)F)F)F)F)F)<#7$F1F6" }}}{PARA 0 "" 0 "" {TEXT -1 178 "Naturally, since we are dealing with \+ sets of summands (the order does not count), we may as well regard the se objects as an increasing sequence of summands that< sum to the size " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 79 ", or equivalently as \" staircases\" with size being the area below the staircase." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "preduce:=proc(e) sort(eval(subs(\{S et=proc() [args] end, Sequence=proc() nargs end\},e))) end:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "rand_part:=draw(IBIU,size=10 0): ureduce(rand_part); preduce(rand_part);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<#7C7$%\"ZGF&F%F%F%F%F%7#F&F'F'F'F'F'F'F'F'F'F'F'F'F'F' F'F'7'F&F&F&F&F&F(F(F(F(F(F(F(73F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&70F& F&F&F&F&F&F&F&F&F&F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7C\"\"\" F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$\"\"#F%F%F%F%F%\"\"&F&F&F&F&F&F&F&\"#9 \"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 " There are much fewer structures than in previous models:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(count(IBIU,size=j),j=0..30);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6A\"\"\"F#\"\"#\"\"$\"\"&\"\"(\"#6\"#: \"#A\"#I\"#U\"#c\"#x\"$,\"=\"$N\"\"$w\"\"$J#\"$(H\"$&Q\"$!\\\"$F'\"$#z \"%-5\"%b7\"%v:\"%e>\"%OC\"%5I\"%=P\"%lX\"%/c" }}}{PARA 0 "" 0 "" {TEXT -1 209 "The random generation process is nontrivial as one must \+ generate objects up to certain symmetries. The first time, counting ta bles are set up on the fly, so that random generation takes a few seco nds for size " }{XPPEDIT 18 0 "n<=100" "1%\"nG\"$+\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "for i from 0 by 20 to 100 \+ do i,preduce(draw(IBIU,size=i)); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$\"\"!%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#?7*\"\"\"F%F%F %F%\"\"$\"\"%\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#S77\"\"\"F%F% F%F%F%F%F%F%F%\"\"#F&F&F&F&F&F&F&\"\"%F'\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#g70\"\"\"F%\"\"#F&F&F&F&\"\"$\"\"%\"\"&F)\"\"'\"#7\" #8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#!)7;\"\"\"\"\"#F&F&F&F&F&F&F& F&F&F&\"\"$F'F'F'F'F'F'F'\"\"%F(\"\"&\"\"(\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"$+\"74\"\"\"F%F%F%F%\"\"$F&F&F&F&F&\"\"&F'F>'\"#5F(\"# 6\"#J" }}}{PARA 0 "" 0 "" {TEXT -1 39 "Next, random generation becomes faster:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "for i to 10 do p reduce(draw(IBIU,size=100)); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#71 \"\"\"F$F$F$F$F$F$F$\"\"#\"\"'\"#5\"#7\"#;\"#A\"#C" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7=\"\"\"F$F$F$F$F$F$F$F$F$F$\"\"#F%F%F%F%F%\"\"$F&F& \"\"%F'\"\"&\"\"*F)\"#<\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#71\"\" \"F$F$F$F$F$\"\"#\"\"$\"\"%F'\"\"*\"#6F)\"#=\"#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#73\"\"\"F$F$F$F$\"\"#F%F%F%F%\"\"%\"\"(F'\"#5\"#7\"#?\" #D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7A\"\"\"F$F$F$F$F$F$F$F$F$F$\"\" #F%F%\"\"$F&F&F&F&F&\"\"%F'F'F'F'F'F'F'F'\"#5\"#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-\"\"%F$F$F$\"\"&\"\"'F&\"\"*\"#=\"#?F)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#78\"\"\"F$F$\"\"#F%F%F%F%F%F%\"\"$\"\"&F'F'F'\" \"'F(F(F(\"\"(\"#9\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7<\"\"\"F$F$ F$F$F$F$F$F$F$F$F$F$F$\"\"#F%F%F%\"\"$\"\"%\"\"&\"\"*\"#5\"#7\"#8\"#A " }}{PARA 11 "" 1 "" {?XPPMATH 20 "6#7<\"\"\"F$F$F$F$F$F$F$F$F$F$F$\"\" #F%F%\"\"$F&\"\"%\"\"&F(F(\"\"(\"\")\"#7F+\"#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7<\"\"\"F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$\"\"%F%\" \"(\"#9\"#B\"#G" }}}{PARA 0 "" 0 "" {TEXT -1 325 "In this particular c ase, the random generation procedure that is automatically built by Co mbstruct coincides with a method especially designed by Wilf for integ er partitions. By design, Combstruct accepts in full generality arbitr ary compositions of Set and Cycle constructions (in addition to Union, Product, Sequence, etc)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 85 "The generating functions are now more complicated \+ since they involve Polya operators." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "gfeqns(op(2..3,IBIU),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/-%\"UG6#%\"zG,&*$,&\"\"\"F,-%\"ZGF'!\"\"F/F,F/F,/-% \"SGF'-%$expG6#-%$SumG6$*&-F&6#)F(&%\"jG6#F,F,F=F//F=;F,%)infinityG/F- F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "gfsolve(op(2..3@,IBIU) ,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%\"ZG6#%\"zGF(/-%\"UGF',$* &F(\"\"\",&!\"\"F.F(F.F0F0/-%\"SGF'-%$expG6#-%$SumG6$,$*()F(&%\"jG6#F. F.,&F " 0 "" {MPLTEXT 1 0 26 "gfseries(op(2..3,IBIU),z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7%/-%\"SG6#%\"zG+5F+\"\" \"\"\"!F-\"\"\"\"\"#\"\"#\"\"$\"\"$\"\"&\"\"%\"\"(\"\"&\"#6\"\"'\"#:\" \"(-%\"OG6#F-\"\")/-%\"UGF*+3F+F-\"\"\"F-\"\"#F-\"\"$F-\"\"%F-\"\"&F- \"\"'F-\"\"(F<\"\")/-%\"ZGF*+%F+F-\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 46 "though they are not otherwise \"known\" to Maple" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "subs(\"\",S(z)); series(\",z=0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#-%$SumG6$,$*()%\"zG&%\"jG6#\" \"\"F0,&F+F0!\"\"F0F2F-F2F2/F-;F0%)infinityG" }}{PARA 8 "" 1 "" {TEXT -1 47 "Error, (in series/exp) unable to compute series" }}}{PARA 0 "" 0 "" {TEXT -1 191 "In such casAes, one has to resort either to simplifi cation by hand (not always possible) or to the literature. Here, it is very well known that the generating function of integer partitions is " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "PP_z:=Product(1/(1-z^k), k=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%PP_zG-%(ProductG 6$*$,&\"\"\"F*)%\"zG%\"kG!\"\"F./F-;F*%)infinityG" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 51 "Order:=12: series(subs(infinity=Order+2,PP_z ),z=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+=%\"zG\"\"\"\"\"!F%\"\"\" \"\"#\"\"#\"\"$\"\"$\"\"&\"\"%\"\"(\"\"&\"#6\"\"'\"#:\"\"(\"#A\"\")\"# I\"\"*\"#U\"#5\"#c\"#6-%\"OG6#F%\"#7" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "Constrained models" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Number of urns in surjections" }}{PARA 0 "" 0 "" {TEXT -1 357 "The approach developed so far may be tuned to analyse a variety parameter s. We explore here the way Combstruct may serve to analyse the number \+ of urns as well as related situations with bounded urn capacBity. We fo cus on counts and building numerical tables. Naturally, random generat ion and exhaustive listing are possible from any of these specificatio ns.\n" }}{PARA 0 "" 0 "" {TEXT -1 180 "We deal here with a fixed numbe r of urns in the model DBDU that corresponds to surjections. Combinato rial specifications may actually be computed in Maple, then used by Co mbstruct." }}{PARA 0 "" 0 "" {TEXT -1 57 "The following procedure comp utes the specifications with " }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 6 " urns." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "surj:=[S,\{S =Sequence(U,card=r),U=Set(Z,card>=1)\}, labelled]: subs(r=5,surj);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%%\"SG<$/%\"UG-%$SetG6$%\"ZG1\"\"\"%% cardG/F$-%)SequenceG6$F'/F.\"\"&%)labelledG" }}}{PARA 0 "" 0 "" {TEXT -1 106 "In passing, this illustrates the use of cardinality modifiers for Sequence, Set, and Cycle constructions." }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 139 "The following counts imply the first few values of the probability distrCibution of the number of urns in a random unconstrai ned surjection:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "for i to 5 do se q(count(subs(r=i,surj),size=m),m=0..10) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-\"\"!\"\"\"F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-\"\"!F#\"\"#\"\"'\"#9\"#I\"#i\"$E\"\"$a#\"$5&\"%A5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6-\"\"!F#F#\"\"'\"#O\"$]\"\"$S&\"%1=\"%' z&\"&]\"=\"&!)f&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-\"\"!F#F#F#\"#C\"$ S#\"%g:\"%+%)\"&C3%\"'!['=\"'?&=)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6- \"\"!F#F#F#F#\"$?\"\"%+=\"&+o\"\"'+g7\"'?T$)\"(+I5&" }}}{PARA 0 "" 0 " " {TEXT -1 67 "and we may build tables of probability distributions au tomatically:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for i to 7 d o seq(evalf(count(subs(r=i,surj),size=m)/count(DBDU,size=m),4),m=0..10 ) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-\"\"!$\"\"\"F#$\"%LL!\"%$\"%# p(!\"&$\"%L8F+$\"%[=!\"'$\"%N@!\"($\"%9@!\")$\"%K=!\"*$\"%69!#5$\"%!y* !#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-\"\"!F#$\"%nm!\"%$D\"%:YF&$\"%n= F&$\"%Xb!\"&$\"%C8F-$\"%kE!\"'$\"%`Y!\"($\"%'>(!\")$\"%&***!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6-\"\"!F#F#$\"%:Y!\"%$\"%+[F&$\"%tFF&$\" %`6F&$\"%>Q!\"&$\"%i5F/$\"%hD!\"'$\"%va!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-\"\"!F#F#F#$\"%+K!\"%$\"%OWF&$\"%JLF&$\"%w$F&$\"%4NF&$\"%)o#F &$\"%2;F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-\"\"!F#F#F#F#F#F#$\"%m5! \"%$\"%&e#F&$\"%&G$F&$\"%)*GF&" }}}{PARA 0 "" 0 "" {TEXT -1 177 "Final ly, we are led to rediscover the corresponding generating functions. U sually, this is done via recurrence computations, and what we obtain h ere is equivalent to the EGF of " }{HYPERLNK 17 "Stirling second kind \+ (partition) numbers" 2 "combinat[stirling2]" "" }{TEXT -1 1 ":" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "for i to 5 do gfsolve(op(2,s ubs(r=i,surj)),labelled,zE) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/- %\"UG6#%\"zG,&-%$expGF'\"\"\"!\"\"F,/-%\"ZGF'F(/-%\"SGF'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%\"UG6#%\"zG,&-%$expGF'\"\"\"!\"\"F,/-% \"ZGF'F(/-%\"SGF',(*$F*\"\"#F,F*!\"#F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%\"UG6#%\"zG,&-%$expGF'\"\"\"!\"\"F,/-%\"ZGF'F(/-%\"SGF',**$ F*\"\"$F,*$F*\"\"#!\"$F*F6F-F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/- %\"UG6#%\"zG,&-%$expGF'\"\"\"!\"\"F,/-%\"ZGF'F(/-%\"SGF',,*$F*\"\"%F,* $F*\"\"$!\"%*$F*\"\"#\"\"'F*F9F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6# <%/-%\"UG6#%\"zG,&-%$expGF'\"\"\"!\"\"F,/-%\"SGF',.*$F*\"\"&F,*$F*\"\" %!\"&*$F*\"\"$\"#5*$F*\"\"#!#5F*F3F-F,/-%\"ZGF'F(" }}}{PARA 0 "" 0 "" {TEXT -1 52 "This suggests a pattern involving Pascal's triangle:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Surj_z:=proc(k) local j; add (binomial(k,j)*(-1)^(k-j)*exp(z)^j,j=0..k) end:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 52 "Surj_z(6); gfsolve(op(2,subs(r=6,surj)),labell ed,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0\"\"\"F$-F%$expG6#%\"zG!\"' *$F%\"\"#\"#:*$F%\"\"$!#?*$F%\"\"%F,*$F%\"\"&F)*$F%\"\"'F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%\"UG6#%\"zG,&-%$expGF'\"\"\"!\"\"F,/-% \"ZGF'F(/-%\"SGF',0F,F,F*!\"'*$F*\"\"#\"#:*$F*\"\"$!#?*$F*\"\"%F8*$F* \"\"&F5*$F*\"\"'F," }}}{PARA 0 "" 0 "" {TEXT -1 55 "For coefficients f inally, we have by straight expansion" }}{EXCHG {PARA 264 "" 0 "" {XPPEDIT 18 0 "Sum(binomial(k,j)*(-1)^(k-j)*j^n,j=0..k)" "-%$SumG6$*(- %)binomialG6$%\"kG%\"jG\"\"\"),$\"\"\"!\"\",&F)F+F*F/F+)F*%\"nGF+/F*; \"\"!F)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 " Surj_nk:=proc(n,k) local j; `if`(n<>0,add(binomial(k,j)*(-1)^(k-j)*j^n ,j=0..k),1) end:" }}}{PARA 0 "" 0 "" {TEXT -1 63 "The formula matches \+ the values obtained by combstruct[gfseries]" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 47 "Order:=16: gfseries(op(2..3,subs(r=7,surj)),z);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7%/-%\"SG6#%\"zG+7F+\"\"\" \"\"(#\"\"(\"\"#\"\")#\"#x\"#7\"\"*#\"#\\\"\"'\"#5#\"%R>\"$S#\"#6#\"%` Z\"$?(\"#7#\"%\\G7\"$q#\"#8#F4\"#F\"#9#\"'6h8\"&+k)\"#:-%\"OG6#F-\"#;/- %\"UGF*+CF+F-\"\"\"#F-F1\"\"##F-F9\"\"$#F-\"#C\"\"%#F-\"$?\"\"\"&#F-FA \"\"'#F-\"%S]\"\"(#F-\"&?.%\"\")#F-\"'!)GO\"\"*#F-\"(+)GO\"#5#F-\")+o \"*R\"#6#F-\"*+;+z%\"#7#F-\"++3-Fi\"#8#F-\",+7Hyr)\"#9#F-\".+!oVn28\"# :FN\"#;/-%\"ZGF*+%F+F-\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "seq(Surj_nk(n,7)/n!,n=0..15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 2\"\"\"\"\"!F$F$F$F$F$F##\"\"(\"\"##\"#x\"#7#\"#\\\"\"'#\"%R>\"$S##\"% `Z\"$?(#\"%\\7\"$q##F)\"#F#\"'6h8\"&+k)" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 262 39 "Number of parts in integer compositions" }}{PARA 0 "" 0 "" {TEXT -1 23 "This is the model IBDU." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "comp_r:=[S,\{S=Sequence(U,card=r),U=Sequence(Z,card>= 1)\},unlabelled]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "for i \+ to 5 do seq(count(subs(r=i,comp_r),size=m),m=0..15) od;" }}{PARA 11 " " 1 "" {XPPMATH 20 "62\"\"!\"\"\"F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }} {PARA 11 "" 1 "" {XPPMATH 20 "62\"\"!F#\"\"\"\"\"#\H"\"$\"\"%\"\"&\"\"' \"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 2\"\"!F#F#\"\"\"\"\"$\"\"'\"#5\"#:\"#@\"#G\"#O\"#X\"#b\"#m\"#y\"#\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "62\"\"!F#F#F#\"\"\"\"\"%\"#5\"#?\"#N\" #c\"#%)\"$?\"\"$l\"\"$?#\"$'G\"$k$" }}{PARA 11 "" 1 "" {XPPMATH 20 "62 \"\"!F#F#F#F#\"\"\"\"\"&\"#:\"#N\"#q\"$E\"\"$5#\"$I$\"$&\\\"$:(\"%,5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "for i to 5 do subs(gfsolv e(op(2,subs(r=i,comp_r)),labelled,z),S(z)) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"zG\"\"\",&!\"\"F&F%F&F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"zG\"\"#,&!\"\"\"\"\"F$F(!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"zG\"\"$,&!\"\"\"\"\"F%F)!\"$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"zG\"\"%,&!\"\"\"\"\"F$F(!\"%" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*&%\"zG\"\"&,&!\"\"\"\"\"F%F)!\"&F(" }}}{PARA 0 "" 0 "" {TEXT -1 115 "The pattern is obvious and this corresponds to an e xplicit (and well-known!) binomial formula for the coefficients." }}} {SECT 1 {PARAI 4 "" 0 "" {TEXT -1 37 "Number of parts in integer partit ions" }}{PARA 0 "" 0 "" {TEXT -1 23 "This is the model IBIU." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "part_r:=[S,\{S=Set(U,card=r) ,U=Sequence(Z,card>=1)\}, unlabelled]:" }}}{PARA 0 "" 0 "" {TEXT -1 32 "We can build tables, like before" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "for i to 5 do seq(count(subs(r=i,part_r),size=m),m=0. .15) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "62\"\"!\"\"\"F$F$F$F$F$F$F$F $F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "62\"\"!F#\"\"\"F$\"\"#F %\"\"$F&\"\"%F'\"\"&F(\"\"'F)\"\"(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 62\"\"!F#F#\"\"\"F$\"\"#\"\"$\"\"%\"\"&\"\"(\"\")\"#5\"#7\"#9\"#;\"#> " }}{PARA 11 "" 1 "" {XPPMATH 20 "62\"\"!F#F#F#\"\"\"F$\"\"#\"\"$\"\"& \"\"'\"\"*\"#6\"#:\"#=\"#B\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "62\"\" !F#F#F#F#\"\"\"F$\"\"#\"\"$\"\"&\"\"(\"#5\"#8\"#=\"#B\"#I" }}}{PARA 0 "" 0 "" {TEXT -1 73 "The generating functions are found by combstruct[ gfsolve] to be rational:" }}{EXCHG {PARA 0 "> " 0 "" J{MPLTEXT 1 0 62 " for i to 5 do gfsolve(op(2,subs(r=i,part_r)),unlabelled,z) od;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%\"ZG6#%\"zGF(/-%\"UGF',$*&F(\"\" \",&!\"\"F.F(F.F0F0/-%\"SGF'F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/- %\"ZG6#%\"zGF(/-%\"UGF',$*&F(\"\"\",&!\"\"F.F(F.F0F0/-%\"SGF'*&F(\"\"# ,**$F(\"\"$F.*$F(F5F0F(F0F.F.F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/ -%\"ZG6#%\"zGF(/-%\"UGF',$*&F(\"\"\",&!\"\"F.F(F.F0F0/-%\"SGF',$*&F(\" \"$,.*$F(\"\"'F.*$F(\"\"&F0*$F(\"\"%F0*$F(\"\"#F.F(F.F0F.F0F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%\"ZG6#%\"zGF(/-%\"UGF',$*&F(\"\" \",&!\"\"F.F(F.F0F0/-%\"SGF'*&F(\"\"%,0*$F(\"#5F.*$F(\"\"*F0*$F(\"\")F 0*$F(\"\"&\"\"#*$F(F?F0F(F0F.F.F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#< %/-%\"ZG6#%\"zGF(/-%\"UGF',$*&F(\"\"\",&!\"\"F.F(F.F0F0/-%\"SGF',$*&F( \"\"&,:*$F(\"#:F.*$F(\"#9F0*$F(\"#8F0*$F(\"#5F.*$F(\"\"*F.*$F(\"\")F.* $F(\"\"(F0*$F(\"\"'F0*$F(F6F0*$F(\"\"#F.F(F.F0F.F0F0" }}}{PARA 0 "" 0 "" {TEXT -1 109 "The form is then easily inferred from the factored re presentations, as oKne recognizes cyclotomic polynomials." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "S5_z:=subs(\",S(z));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%S5_zG,$*&%\"zG\"\"&,:*$F'\"#:\"\"\"*$F'\"#9! \"\"*$F'\"#8F/*$F'\"#5F,*$F'\"\"*F,*$F'\"\")F,*$F'\"\"(F/*$F'\"\"'F/*$ F'F(F/*$F'\"\"#F,F'F,F/F,F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "factor(S5_z); series(\",z=0,20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*.%\"zG\"\"&,&!\"\"\"\"\"F%F)!\"&,&F%F)F)F)!\"#,(*$F%\"\"#F)F% F)F)F)F(,&F.F)F)F)F(,,*$F%\"\"%F)*$F%\"\"$F)F.F)F%F)F)F)F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+C%\"zG\"\"\"\"\"&F%\"\"'\"\"#\"\"(\"\"$\"\" )\"\"&\"\"*\"\"(\"#5\"#5\"#6\"#8\"#7\"#=\"#8\"#B\"#9\"#I\"#:\"#P\"#;\" #Z\"#<\"#d\"#=\"#q\"#>-%\"OG6#F%\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "z^5/mul(1-z^i,i=1..5); factor(\"); series(\",z=0,20); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*.%\"zG\"\"&,&\"\"\"F'F$!\"\"F(,&F 'F'*$F$\"\"#F(F(,&F'F'*$F$\"\"$F(F(,&F'F'*$F$\"\"%F(F(,&F'F'*$F$F%F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*.%\"zG\"\"&,&!\"\L"\"\"\"F%F)!\" &,&F%F)F)F)!\"#,(*$F%\"\"#F)F%F)F)F)F(,&F.F)F)F)F(,,*$F%\"\"%F)*$F%\" \"$F)F.F)F%F)F)F)F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+C%\"zG\"\"\" \"\"&F%\"\"'\"\"#\"\"(\"\"$\"\")\"\"&\"\"*\"\"(\"#5\"#5\"#6\"#8\"#7\"# =\"#8\"#B\"#9\"#I\"#:\"#P\"#;\"#Z\"#<\"#d\"#=\"#q\"#>-%\"OG6#F%\"#?" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "Bounded capacity in the DBDU model (hashing)" }}{PARA 0 "" 0 "" {TEXT -1 74 "We now consider configurations where the number of urns i s a fixed number " }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 118 ", so t hat one can relax the constraint that urns need to be nonempty. We are thus dealing with the collection of the " }{XPPEDIT 18 0 "r^n" ")%\" rG%\"nG" }{TEXT -1 14 " functions of " }{XPPEDIT 18 0 "[1..r]" "7#;\" \"\"%\"rG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "[1..r]" "7#;\"\"\"%\"rG " }{TEXT -1 121 ", not necessarily surjections. Such specifications al so describe words (size being length) on an alphabet of cardinality " }{XPPEDIT 18 0 "r"M "I\"rG6\"" }{TEXT -1 70 " and this may be used to m odel hashing sequences in a table of length " }{XPPEDIT 18 0 "r" "I\"r G6\"" }{TEXT -1 37 ". The set of all possible sequences (" }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 24 " fixed) is specified by" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "hash_r:=[S,\{S=Prod(U$r),U=Set(Z)\} ,labelled]: subs(r=10,hash_r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%% \"SG<$/F$-%%ProdG6#-%\"$G6$%\"UG\"#5/F--%$SetG6#%\"ZG%)labelledG" }}} {PARA 0 "" 0 "" {TEXT -1 26 "Here, we use a hack, with " }{HYPERLNK 17 "Prod" 2 "combstruct[specification]" "" }{TEXT -1 222 " instead of \+ \"Sequence\" with fixed cardinality, since the current version of Comb struct does not accept Sequence applied to an argument that leads to a n Epsilon structure, even in the case of a bounded cardinality modifie r." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "We change the reduction proc edure to take this new construct into account." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 196 "lreduce:=proc(e) eval(Nsubs(\{Set=proc() \{arg s\} end, Prod=proc() [args] end\},e)) end:\nureduce:=proc(e) eval(subs (\{Set=proc() \{[args]\} end, Sequence=proc() [args] end,Prod=proc() [ args] end\},e)) end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The numbe r of objects of size " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 19 " is , as predicted, " }{XPPEDIT 18 0 "10^n" ")\"#5%\"nG" }{TEXT -1 1 "." } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "seq(count(subs(r=10,hash_r),size=j ),j=0..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-\"\"\"\"#5\"$+\"\"%+5\" &++\"\"'++5\"(+++\"\")+++5\"*++++\"\"+++++5\",+++++\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Next, we consider ur ns with a bounded maximum capacity " }{XPPEDIT 18 0 "b" "I\"bG6\"" } {TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "hash_br:=[S ,\{S=Prod(U$r),U=Set(Z,card<=b)\}, labelled]: subs(\{r=5,b=3\},hash_br );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%%\"SG<$/F$-%%ProdG6#-%\"$G6$% \"UG\"\"&/F--%$SetG6$%\"ZG1%%cardG\"\"$%)labelledG" }}}{PARA 0 "" 0 " " {TEXTO -1 90 "Such specifications also describe words (size being len gth) on an alphabet of cardinality " }{XPPEDIT 18 0 "r" "I\"rG6\"" } {TEXT -1 41 ", given that no letter is used more than " }{XPPEDIT 18 0 "b" "I\"bG6\"" }{TEXT -1 59 " times. This models hashing sequences i n a table of length " }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 26 " whe n pages have capacity " }{XPPEDIT 18 0 "b" "I\"bG6\"" }{TEXT -1 2 ". \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "lreduce(draw(subs(r=10,h ash_r),size=30));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,<$&%\"ZG6#\"#D& F&6#\"#G<'&F&6#\"\"*&F&6#\"#<&F&6#\"\"&&F&6#\"#I&F&6#\"#B<%&F&6#\"#H&F &6#\"\"'&F&6#\"#8<#&F&6#\"\"(<'&F&6#\"#9&F&6#\"\"\"&F&6#\"#=&F&6#\"#C& F&6#\"#A<%&F&6#\"#@&F&6#\"#5&F&6#\"#6%(EpsilonG<'&F&6#\"#>&F&6#\"\"#&F &6#\"#;&F&6#\"#E&F&6#\"#F<%&F&6#\"\"%&F&6#\"#?&F&6#\"#7<%&F&6#\"\"$&F& 6#\"#:&F&6#\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "lreduce (draw(subs(\{r=10,b=4\},hash_br),size=30));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,<%&%\"ZG6#\"#9&F&6#\"\"*P&F&6#\"#C<&&F&6#\"#>&F&6#\"\" &&F&6#\"#A&F&6#\"#:<&&F&6#\"\"#&F&6#\"\"'&F&6#\"#;&F&6#\"#F<$&F&6#\"#B &F&6#\"#E<$&F&6#\"#H&F&6#\"#D<&&F&6#\"#@&F&6#\"\"(&F&6#\"#I&F&6#\"#=<% &F&6#\"\"\"&F&6#\"\"%&F&6#\"#<<$&F&6#\"\"$&F&6#\"#7<%&F&6#\"#5&F&6#\"# 6&F&6#\"#?<%&F&6#\"#G&F&6#\"#8&F&6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 82 "The following command automatically creates a table of maximum urn occupancy: the " }{XPPEDIT 18 0 "j" "I\"jG6\"" }{TEXT -1 17 "-th entr y in the " }{XPPEDIT 18 0 "b" "I\"bG6\"" }{TEXT -1 33 "-th line is the probability that " }{XPPEDIT 18 0 "j" "I\"jG6\"" }{TEXT -1 53 " balls thrown into 10 urns fit into urns of capacity " }{XPPEDIT 18 0 "b" "I \"bG6\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 " for i 1 to 6 do seq(evalf(count(subs(\{r=10,b=i\},hash_br),size=j)/cou nt(subs(r=10,hash_r),size=j),4),j=0..20); od;" }}{PARA 8 "" 1 "" {TEXT -1 31 "Syntax error, unexpected number" }}}{PARA 0 "" 0 "" {TEXT -1 75 "Naturally, Combstruct automatically recognizes \"impossib le\" cQonfigurations:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "draw (subs(\{r=10,b=3\},hash_br),size=31);" }}{PARA 8 "" 1 "" {TEXT -1 69 " Error, (in combstruct/drawgrammar) there is no structure of this size " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "Bounded capacity in the IBD U model (bosons)" }}{PARA 0 "" 0 "" {TEXT -1 161 "This is a model of i nteger compositions. (A more sophisticated example that is related to \+ submarine detection is treated below.) We consider here a fixed number " }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 139 " of urns and this is e xactly the so-called \"Bose-Einstein\" model of statistical physics, w here the corresponding objects are called bosons." }}{PARA 0 "" 0 "" {TEXT -1 84 "Mark Kobrak, a chemist at the University of Chicago wrote to us (December 13, 1996):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 535 " The basic problem is this: I am inte rested in simulating a problem in laser spectroscopy. A molecule has \+ n modes, and I need to generateR every possible combination which place s up to m quanta in each mode. As I go through each one, I will analy ze its energy.\n This is exactly like a problem where, given n \+ jars, you may put up to m marbles in each jar. The marbles are indist inguishable. I know how to count the number of combinations, but I nee d a good way to program a computer to go through all these combination s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "We \+ let again " }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 31 " denote the nu mber of urns and " }{XPPEDIT 18 0 "b" "I\"bG6\"" }{TEXT -1 130 " the b ucket capacity, i.e., the maximum number of urns that may fit into any given urn. We have the model of integer compositions:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "boson_br:=[S,\{S=Prod(U$r),U=Sequen ce(Z,card<=b)\}, unlabelled]: subs(\{r=5,b=3\},boson_br);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%%\"SG<$/%\"UG-%)SequenceG6$%\"ZG1%%cardG\"\"$ /F$-%%ProdG6#-%\"$G6$F'\"\"&%+unlabelledG" }}}{PARA 0 "" 0S "" {TEXT -1 59 "For instance, here are all the 95 possible ways of putting " } {XPPEDIT 18 0 "j" "I\"jG6\"" }{TEXT -1 10 " marbles (" }{XPPEDIT 18 0 "j<=4" "1%\"jG\"\"%" }{TEXT -1 7 ") into " }{XPPEDIT 18 0 "r=5" "/%\"r G\"\"&" }{TEXT -1 42 " jars, each jar being of maximum capacity " } {XPPEDIT 18 0 "b=2" "/%\"bG\"\"#" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 51 "seq(count(subs(\{r=5,b=2\},boson_br),size=j),j =1..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"&\"#:\"#I\"#X" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "for j to 4 do j=map(ureduce, allstructs(subs(\{r=5,b=2\},boson_br),size=j)) od;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/\"\"\"7'7'7#%\"ZG%(EpsilonGF)F)F)7'F)F)F)F'F)7'F)F)F 'F)F)7'F)F)F)F)F'7'F)F'F)F)F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/\"\" #717'7#%\"ZG%(EpsilonGF)F'F)7'F)F)7$F(F(F)F)7'F)F'F'F)F)7'F)F)F)F+F)7' F)F)F)F)F+7'F)F'F)F)F'7'F'F)F'F)F)7'F)F)F)F'F'7'F)F+F)F)F)7'F+F)F)F)F) 7'F)F'F)F'F)7'F'F'F)F)F)7'F)F)F'F)F'7'F)F)F'F'F)7'F'F)F)F)F'" }}{PARA 12 "" 1 "" {XPPMATH 20 T"6#/\"\"$7@7'7#%\"ZG7$F(F(%(EpsilonGF*F*7'F*F*F 'F'F'7'F'F*F*F'F'7'F'F*F'F*F'7'F'F'F*F'F*7'F*F)F*F*F'7'F*F)F*F'F*7'F)F *F'F*F*7'F*F'F'F'F*7'F*F*F'F)F*7'F'F*F)F*F*7'F*F'F)F*F*7'F*F*F)F*F'7'F )F'F*F*F*7'F)F*F*F*F'7'F*F'F*F*F)7'F'F*F*F)F*7'F'F*F*F*F)7'F*F)F'F*F*7 'F*F'F*F)F*7'F'F'F'F*F*7'F*F*F'F*F)7'F)F*F*F'F*7'F*F'F'F*F'7'F'F'F*F*F '7'F*F*F*F)F'7'F*F'F*F'F'7'F'F*F'F'F*7'F*F*F*F'F)7'F*F*F)F'F*" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/\"\"%7O7'7$%\"ZGF(%(EpsilonG7#F(F)F*7 'F)F'F)F)F'7'F*F*F'F)F)7'F)F)F*F*F'7'F*F'F*F)F)7'F*F)F*F'F)7'F)F'F*F*F )7'F)F)F'F'F)7'F)F'F)F*F*7'F'F)F*F*F)7'F*F*F)F)F'7'F)F'F*F)F*7'F'F*F)F )F*7'F)F)F)F'F'7'F)F)F'F*F*7'F)F*F'F*F)7'F*F)F*F*F*7'F*F)F'F)F*7'F)F*F *F)F'7'F)F'F)F'F)7'F*F'F)F*F)7'F)F)F*F'F*7'F'F)F)F'F)7'F*F*F*F*F)7'F*F )F)F'F*7'F)F*F)F*F'7'F*F*F)F'F)7'F'F)F'F)F)7'F)F*F*F'F)7'F'F*F*F)F)7'F *F*F*F)F*7'F*F'F)F)F*7'F'F'F)F)F)7'F*F)F'F*F)7'F*F*F)F*F*7'F)F)F'F)F'7 'F*F)F*F)F'7'F)F*F'F)F*7'F)F'F'F)F)7'F)F*F)F'F*7'F'F)F)F*F*7'F)F*F*F*F *7'F'F*F)F*F)7'F'F)F)F)F'7'F*F)F)F*F'" }}}}}{SECT 1 {PUARA 3 "" 0 "" {TEXT -1 17 "Stack polyominoes" }}{PARA 0 "" 0 "" {TEXT -1 84 "Polyomi noes are familiar objects of combinatorial mathematics. A stack polyom ino or " }{TEXT 261 5 "stack" }{TEXT -1 245 " is a piling up of nonemp ty integer intervals, each interval being included in the previous one . The number of intervals comprising a stack are called the height; th e total length of the intervals is called the size. For instance, the \+ collection" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "[1,12],[3,8],[ 4,7],[4,7],[6,7];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'7$\"\"\"\"#77$\" \"$\"\")7$\"\"%\"\"(F)7$\"\"'F+" }}}{PARA 0 "" 0 "" {TEXT -1 31 "is a \+ stack of height 5 and size" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "(12-1)+(8-3)+(7-4)+(7-4)+(7-6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"#B" }}}{PARA 0 "" 0 "" {TEXT -1 83 "We may assume stacks to be left justified, starting at 1. Stacks of height exactly " }{XPPEDIT 18 0 " r" "I\"rG6\"" }{TEXT -1 19 " are specified by (" }{XPPEDIT 18 0 "r=5" "/%\"rG\"\V"&" }{TEXT -1 16 " in the example)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "stack:=[st,\{st=Prod(left,right),left=Set(U,card= r),right=Set(U,card=1)\},unlabelled]: subs(r=5,s tack);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%%#stG<&/%%leftG-%$SetG6$% \"UG/%%cardG\"\"&/%&rightG-F)6$F+2F-F./F$-%%ProdG6$F'F0/F+-%)SequenceG 6$%\"ZG1\"\"\"F-%+unlabelledG" }}}{PARA 0 "" 0 "" {TEXT -1 85 "Combina torially, we view a stack as an ascending staircase (the left part) of height " }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 71 " followed by a d escending staircase (the right part) of height at most " }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 77 ". Staircases of fixed height are define d as partitions into bounded summands." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 57 "The distribution of height in stacks \+ is for small height:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "for \+ i to 6 do seq(count(subs(r=i,stack),size=m),m=0..20) od;" }}{PARA 11 " " 1 "" {XPPMATH 20 "67\"\"!\"\"\"WF$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $" }}{PARA 11 "" 1 "" {XPPMATH 20 "67\"\"!F#\"\"\"\"\"#\"\"%\"\"'\"\"* \"#7\"#;\"#?\"#D\"#I\"#O\"#U\"#\\\"#c\"#k\"#s\"#\")\"#!*\"$+\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "67\"\"!F#F#\"\"\"\"\"#\"\"&\"\"*\"#;\"#D \"#R\"#c\"#!)\"$4\"\"$Z\"\"$#>\"$\\#\"$:$\"$'R\"$*[\"$+'\"$E(" }} {PARA 11 "" 1 "" {XPPMATH 20 "67\"\"!F#F#F#\"\"\"\"\"#\"\"&\"#5\"#>\"# K\"#a\"#%)\"$H\"\"$!>\"$v#\"$'Q\"$O&\"$E(\"$t*\"%#G\"\"%s;" }}{PARA 11 "" 1 "" {XPPMATH 20 "67\"\"!F#F#F#F#\"\"\"\"\"#\"\"&\"#5\"#?\"#N\"# h\"#**\"$f\"\"$W#\"$p$\"$T&\"$%y\"%46\"%^:\"%J@" }}{PARA 11 "" 1 "" {XPPMATH 20 "67\"\"!F#F#F#F#F#\"\"\"\"\"#\"\"&\"#5\"#?\"#O\"#k\"$1\"\" $u\"\"$u#\"$D%\"$S'\"$_*\"%%Q\"\"%*)>" }}}{PARA 0 "" 0 "" {TEXT -1 30 "The total number of stacks is:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "for i to 6 do for m from 0 to 10 do stnum[i,m]:=count(subs(r= i,stack),size=m) od od: for m to 6 do m,convert([seq(stnum[i,m],i=1..6 )],`+`) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"F#" }}{PARA 11 " " 1X "" {XPPMATH 20 "6$\"\"#F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"$ \"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"%\"\")" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$\"\"&\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"'\"# F" }}}{PARA 0 "" 0 "" {TEXT -1 8 "This is " }{TEXT 272 5 "M1102" } {TEXT -1 8 " of the " }{TEXT 263 33 "Encyclopedia of Integer Sequences " }{TEXT -1 161 ". The table given there is incomplete. However, from \+ an earlier computation of generating functions, we have a formula for \+ the generating function of all stacks:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "Q:=Product(1-z^k,k=1..n); Stack_z:=1+Sum(z^r/subs(n=r ,Q)/subs(n=r-1,Q),r=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"QG-%(ProductG6$,&\"\"\"F))%\"zG%\"kG!\"\"/F,;F)%\"nG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%(Stack_zG,&\"\"\"F&-%$SumG6$*()%\"zG%\"rGF&-%( ProductG6$,&F&F&)F,%\"kG!\"\"/F3;F&F-F4-F/6$F1/F3;F&,&F-F&F4F&F4/F-;F& %)infinityGF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Order:=30: series(value(subs(infinity=OrderY+3,Stack_z)),z=0);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#+[o%\"zG\"\"\"\"\"!F%\"\"\"\"\"#\"\"#\"\"%\"\"$\"\") \"\"%\"#:\"\"&\"#F\"\"'\"#Z\"\"(\"#z\"\")\"$I\"\"\"*\"$4#\"#5\"$I$\"#6 \"$7&\"#7\"$%y\"#8\"%$=\"\"#9\"%l<\"#:\"%/E\"#;\"%/Q\"#<\"%/b\"#=\"%)* y\"#>\"&S7\"\"#?\"&!)e\"\"#@\"&xA#\"#A\"&[5$\"#B\"&.I%\"#C\"&?#f\"#D\" &)4\")\"#E\"'%[5\"\"#F\"'p(\\\"\"#G\"'q??\"#H-%\"OG6#F%\"#I" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "A problem in submarine detection " }}{PARA 0 "" 0 "" {TEXT -1 50 "Problem 68-16 that appeared in the 19 68 volume of " }{TEXT 264 11 "SIAM Review" }{TEXT -1 18 " reads as fol lows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 79 "Problem 68-16. A combinatorial Problem, by Melda Hayes (Ocean Tech nology Inc.)." }}{PARA 260 "" 0 "" {TEXT -1 21 "In how many ways can \+ " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 35 " identical balls be dist ributed in " }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 137 " boxes in a \+ row such that each pair of adjacent boxes contains at least 4 baZlls? \+ This problem arose in some work on submarine detection." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 244 "The problem is thus relative to integer compositions with constrained summands. The const raints are reminiscent of maximum capacity problems but they concern s uccessive summands. For pedagogical reasons, we decompose the solution in two phases:" }}{PARA 16 "" 0 "" {TEXT -1 94 "a problem of counting words over a 5-letter alphabet that translates the succession contrai nt;" }}{PARA 16 "" 0 "" {TEXT -1 21 "the original problem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "The word \+ counting problem" }}{PARA 0 "" 0 "" {TEXT -1 39 "Consider an alphabet \+ comprising letters" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A=seq( a.j,j=0..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG6'%#a0G%#a1G%#a2 G%#a3G%#a4G" }}}{PARA 0 "" 0 "" {TEXT -1 6 "There " }{XPPEDIT 18 0 "a. j" "(%\"aG%\"jG" }{TEXT -1 6 ", for " }{XPPEDIT 18 0 "j<4" "2%\"jG\"\" %" }{TEXT -1 4[3 " represents symbolically a summand of size " } {XPPEDIT 18 0 "j" "I\"jG6\"" }{TEXT -1 15 "; the quantity " }{XPPEDIT 18 0 "a4" "I#a4G6\"" }{TEXT -1 37 " represents a summand of cardinalit y " }{TEXT 273 8 "at least" }{TEXT -1 56 " 4. The first grammar specif ies words over the alphabet " }{XPPEDIT 18 0 "A" "I\"AG6\"" }{TEXT -1 75 " such that the sum of indices of any two consecutive letters is at least 4." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "grammar1:=[Q4, \{\n seq(Q.i=Union(Epsilon,seq(Prod(a.j,Q.j),j=4-i..4)),i=0..4),\n \+ seq(a.j=Z,j=0..4)\n\},unlabelled];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)grammar1G7%%#Q4G<,/%#Q0G-%&UnionG6$%(EpsilonG-%%ProdG6$%#a4GF&/% #Q1G-F+6%F--F/6$%#a3G%#Q3GF./%#Q2G-F+6&F--F/6$%#a2GF;F6F./%#a1G%\"ZG/F 8FC/F@FC/F1FC/%#a0GFC/F&-F+6(F--F/6$FHF)-F/6$FBF3F>F6F./F9-F+6'F-FNF>F 6F.%+unlabelledG" }}}{PARA 0 "" 0 "" {TEXT -1 159 "The grammar above i s just a translation of the finite automaton that recognizes the langu age of symbolic constraints. We can solve the\ counting problem easily. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "seq(count(grammar1,size= j),j=0..20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "67\"\"\"\"\"&\"#:\"#b\"$ !>\"$r'\"%`B\"%s#)\"&c!H\"'\"4-\"\"'r'e$\"(V,E\"\"(%HFW\")#fab\"\")1&[ Y&\"*Y')*>>\"*Pfbu'\"+FC%*pB\"+%f1kK)\",vJZ`#H\"-3BJxF5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "gfsolve(op(2..3,grammar1),z);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#<-/-%\"ZG6#%\"zGF(/-%#Q4GF',$*&,,*$F( \"\"$!\"\"\"\"\"F2F(\"\"#*$F(\"\"%F2*$F(F3!\"$F2,.F1F2F6F0F4F1F/!\"%F( F0*$F(\"\"&F2F1F1/-%#a3GF'F(/-%#a4GF'F(/-%#Q3GF'*&,(F1F2F(F1F6F2F2F8F1 /-%#a2GF'F(/-%#Q2GF',$*$F8F1F1/-%#a0GF'F(/-%#Q0GF',$*&,*F2F2F6F1F/F2F( !\"#F2F8F1F1/-%#a1GF'F(/-%#Q1GF'*&,&F1F2F(F2F2F8F1" }}}{PARA 0 "" 0 " " {TEXT -1 57 "The generating function of all words has been found to \+ be" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Q_z:=factor(subs(\",Q4 (z)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Q_zG,$*(,&!\"\"\"\"\"%\"z GF)F),(*$F*\"\"$F)F*!\"$F(F)F),.F(F)*$F*\"\"#F-*$F*\"\"%F(F,!\"%F*F-*$ F*\"\]"&F)F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "series(Q_z ,z=0,20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+M%\"zG\"\"\"\"\"!\"\"&\" \"\"\"#:\"\"#\"#b\"\"$\"$!>\"\"%\"$r'\"\"&\"%`B\"\"'\"%s#)\"\"(\"&c!H \"\")\"'\"4-\"\"\"*\"'r'e$\"#5\"(V,E\"\"#6\"(%HFW\"#7\")#fab\"\"#8\")1 &[Y&\"#9\"*Y')*>>\"#:\"*Pfbu'\"#;\"+FC%*pB\"#<\"+%f1kK)\"#=\",vJZ`#H\" #>-%\"OG6#F%\"#?" }}}{PARA 0 "" 0 "" {TEXT -1 56 "The asymptotics resu lts from locating the dominant pole:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Digits:=30: fsolve(denom(Q_z),z);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6'$!?%yHBOsPBOic12Do\"!#H$!?f%)H[&e5&Gx.g-I3$)!#I$\"?Ls L&e()3Gqlaw'HYGF($\"?$\\9]Q6G,d!*yY@(48F%$\"?89ht!yzZ**Gs%f)*=>F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "rho:=fsolve(denom(Q_z),z,0.. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoG$\"?LsL&e()3Gqlaw'HYG!#I " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "c:=-subs(z=rho,numer(Q_ z)/diff(denom(Q_z),z))/rho;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG$ \"?jJPYw%zOj,\"*p, " 0 "" {MPLTEXT 1 0 42 "Q_n_asympt:=proc(n) round(c*rho^(-n)) end:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 53 "count(grammar1,size=30); Q_n_asympt(30); evalf (\"/\"\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"2\\,V*G*o]%H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"2V-V*G*o]%H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?t#R*)ex\">.++++++5!#H" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "The integer composition problem" }}{PARA 0 "" 0 "" {TEXT -1 52 "Fo r the original problem, we only need to interpret " }{XPPEDIT 18 0 "a. j" "(%\"aG%\"jG" }{TEXT -1 0 "" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "j< 4" "2%\"jG\"\"%" }{TEXT -1 0 "" }{TEXT -1 31 " as meaning a summand of value " }{XPPEDIT 18 0 "j" "I\"jG6\"" }{TEXT -1 0 "" }{TEXT -1 17 ", \+ that is to say " }{XPPEDIT 18 0 "Z" "I\"ZG6\"" }{TEXT -1 0 "" }{TEXT -1 10 " repeated " }{XPPEDIT 18 0 "j" "I\"jG6\"" }{TEXT -1 0 "" } {TEXT -1 12 " times, and " }{XPPEDIT 18 0 "a4" "I#a4G6\"" }{TEXT -1 0 "" }{TEXT -1 34 " as a summand of value at least 4." }}{EXCHG {PARA 0 _ "> " 0 "" {MPLTEXT 1 0 156 "grammar2:=[Q4,\{\n seq(Q.i=Union(Epsilo n,seq(Prod(a.j,Q.j),j=4-i..4)),i=0..4),\n seq(a.j=Sequence(Z,card=j ),j=0..3),a4=Sequence(Z,card>=4)\n\},unlabelled];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)grammar2G7%%#Q4G<,/%#Q0G-%&UnionG6$%(EpsilonG-%%Prod G6$%#a4GF&/F1-%)SequenceG6$%\"ZG1\"\"%%%cardG/%#a0G-F46$F6/F9\"\"!/%#a 3G-F46$F6/F9\"\"$/%#a2G-F46$F6/F9\"\"#/%#a1G-F46$F6/F9\"\"\"/%#Q1G-F+6 %F--F/6$FA%#Q3GF./%#Q2G-F+6&F--F/6$FGFZFVF./F&-F+6(F--F/6$F;F)-F/6$FMF SFgnFVF./FX-F+6'F-F^oFgnFVF.%+unlabelledG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 36 "seq(count(grammar2,size=j),j=0..30);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6A\"\"#\"\"\"F$F$\"\"(\"#6\"#<\"#D\"#^\"#%*\"$l\" \"$!G\"$'\\\"$())\"%w:\"%qF\"%!)[\"%I')\"&w_\"\"&!*p#\"&cw%\"&$=%)\"' \"y[\"\"'@HE\"'GXY\"'*p?)\"(\"4]9\"(]Ac#\"(ss_%\"(/\"**z\")cM89" }}} {PARA 0 "" 0 "" {TEXT -1 46 "We then get the generating function as be fore." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "gr2_sys:=gfsolve(op (2..3,grammar2),z);" }}{PARA 12` "" 1 "" {XPPMATH 20 "6#>%(gr2_sysG<-/- %\"ZG6#%\"zGF*/-%#a0GF)\"\"\"/-%#a4GF),$*&F*\"\"%,&!\"\"F.F*F.F6F6/-%# Q4GF)*&,.*$F*F4F6*$F*\"\"#F6F>F.*$F*\"\"'F.*$F*\"\"$!\"#F*F.F.,2*$F*\" \"*F.*$F*\"\")F.F?F6*$F*\"\"&!\"$F " 0 "" {MPLTEXT 1 0 28 "QQ_z:=factor( subs(\",Q4(z)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%QQ_zG**,&!\"\" \"\"\"%\"zGF(F(,,*$F)\"\"&F(*$F)\"\"%F(*$F)\"\"#!\"#F)!\"$F1F(F(,&F)F( F(F(F',.*$F)\"\")F(F+F'F-F1*$F)\"\"$F'F)F'F(F(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "series(QQ_z,z=0,31);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+]o%\"zG\"\"#\"\"!\"\"\"\"\"\"F'\"\"#F'\"\"$\"\"(\"\"% \"#6\"\"&\"#<\"\"'\"#D\"\"(\"#^\"\")\"#%*\"\"*\"$l\"\"#5\"$!G\"#6\"$' \\\"#7\"$())\"#8\"%w:\"#9\"%qF\"#:\"%!)[\"#;\"%I')\"#<\"&w_\"\"#=\"&!* p#\"#>\"&cw%\"#?\"&$=%)\"#@\"'\"y[\"\"#A\"'@HE\"#B\"'GXY\"#C\"'*ap?)\"# D\"(\"4]9\"#E\"(]Ac#\"#F\"(ss_%\"#G\"(/\"**z\"#H\")cM89\"#I-%\"OG6#F' \"#J" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "rho:=fsolve(denom(Q Q_z),z,0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoG$\"?2Rf\"yEWN^ 2NW\\'fc!#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "QQ_n_asympt :=proc(n) round(subs(z=rho,-1/diff(denom(QQ_z),z)/rho*numer(QQ_z))*rho ^(-n)) end: QQ_n_asympt(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&roun dG6#,$)$\"?2Rf\"yEWN^2NW\\'fc!#I,$%\"nG!\"\"$\"?z@QtsL*oJg`\"3A>aF*" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "count(grammar2,size=40); Q Q_n_asympt(40); evalf(\"\"/\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"+ '[l8>%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"+CeO\">%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?]VugqJ$HH!e$>*******!#I" }}}{PARA 0 "" 0 "" {TEXT -1 148 "This solves the original problem. A detailed solution in volving reduction of infinite matrices was submitted by D. R. Breach, \+ University of Toronto." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "Fixing the number of boxes" b}}{PARA 0 "" 0 "" {TEXT -1 96 "The published sol ution by D. R. Breach also provided detailed tables for number of ball s (size) " }{XPPEDIT 18 0 "n<=20" "1%\"nG\"#?" }{TEXT -1 0 "" }{TEXT -1 21 " and number of boxes " }{XPPEDIT 18 0 "r<=10" "1%\"rG\"#5" } {TEXT -1 0 "" }{TEXT -1 72 ". Like before, we could generate specifica tions for each given value of " }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 0 "" }{TEXT -1 50 ". However, it is possible to solve simultaneousl y " }{TEXT 265 3 "all" }{TEXT -1 180 " such problems by means of biva riate generating functions. Roughly, Combstruct makes it possible to i nsert marks (in the form of Epsilons that do not modify the counting r esults)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "The specification " }{XPPEDIT 18 0 "grammar1" "I)grammar1G6\"" } {TEXT -1 0 "" }{TEXT -1 60 " generates objects where all atoms are ano nymously labelled " }{XPPEDIT 18 0 "Z" "I\"ZG6\"" }{TEXT -1 0 "" } {TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "acllstructs( grammar1,size=2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#71-%%ProdG6$%\"ZG -F%6$F'%(EpsilonGF$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 230 "If we wish to keep track of additional informations, we \+ can just take products with suitable structures of size 0 (Epsilons). \+ This is a general programming technique of Combstruct. Here we use b. j to mark an occurrence of an a.j." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "grammar1bis:=[Q4,\{\n seq(Q.i=Union(Epsilon,seq(P rod(a.j,Q.j),j=4-i..4)),i=0..4),\n seq(a.j=Prod(Z,b.j),j=0..4),\n \+ seq(b.j=Epsilon,j=0..4)\n\},unlabelled];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,grammar1bisG7%%#Q4G<1/%#Q0G-%&UnionG6$%(EpsilonG-%%P rodG6$%#a4GF&/%#Q1G-F+6%F--F/6$%#a3G%#Q3GF./%#Q2G-F+6&F--F/6$%#a2GF;F6 F./F&-F+6(F--F/6$%#a0GF)-F/6$%#a1GF3F>F6F./F9-F+6'F-FGF>F6F./%#b1GF-/% #b2GF-/%#b3GF-/%#b4GF-/%#b0GF-/FF-F/6$%\"ZGFV/FI-F/6$FZFN/F@-F/6$FZFP/ F8-F/6$FZFR/F1-F/6$FZFT%+unlabelledG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "subs(\{Prod=``d,Epsilon=NULL\},allstructs(grammar1bis, size=2));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#71-%!G6$-F%6$%\"ZG%#b3G-F %6#F'-F%6$-F%6$F)%#b2GF+-F%6$-F%6$F)%#b4G-F%6#F/-F%6$F4-F%6#F4-F%6$F/F ;-F%6$F4-F%6#-F%6$F)%#b1G-F%6$F4-F%6#-F%6$F)%#b0G-F%6$F'FA-F%6$F'F;-F% 6$FCF;-F%6$FCF+-F%6$F/F7-F%6$FJF;-F%6$F'F7-F%6$F4F+" }}}{PARA 0 "" 0 " " {TEXT -1 41 "Likewise, we may insert a generic marker " }{XPPEDIT 18 0 "b" "I\"bG6\"" }{TEXT -1 2 " f" }{TEXT -1 49 "or each summand in \+ the compositions described by " }{XPPEDIT 18 0 "grammar2" "I)grammar2G 6\"" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "gram mar2bis:=[Q4,\{\n seq(Q.i=Union(Epsilon,seq(Prod(a.j,Q.j),j=4-i..4) ),i=0..4),\n seq(a.j=Prod(b,Sequence(Z,card=j)),j=0..3),a4=Prod(b,S equence(Z,card>=4)),\n b=Epsilon\n\},unlabelled];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,grammar2bisG7%%#Q4G<-/%#Q0G-%&UnionG6$%(EpsilonG- %%ProdG6$%#a4GF&/%#a0G-F/6$%\"bG-%)SequenceG6$%\"ZG/%%cardG\"\"!/F1-F/ 6$F6-F86$F:1\"\"%F " 0 "" {MPLTEXT 1 0 39 "gfeqns(op(2..3,grammar2bis),z,[[u,b ]]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7./-%#Q4G6$%\"zG%\"uG,.\"\"\"F +*&-%#a0GF'F+-%#Q0GF'F+F+*&-%#a1GF'F+-%#Q1GF'F+F+*&-%#a2GF'F+-%#Q2GF'F +F+*&-%#a3GF'F+-%#Q3GF'F+F+*&-%#a4GF'F+F%F+F+/F/,&F+F+F@F+/F>,,F+F+F1F +F6F+F;F+F@F+/F4,(F+F+F;F+F@F+/F2*&-%\"bGF'F+-%\"ZGF'F+/FKF)/F7*&FKF+F M\"\"#/FA*&FKF+,,*$,&F+F+FM!\"\"FXF+FXF+FMFX*$FMFRFX*$FM\"\"$FXF+/F9,* F+F+F6F+F;F+F@F+/FMF(f/F<*&FKF+FMFen/F-FK" }}}{PARA 261 "" 0 "" {TEXT 266 12 "Solving for " }{XPPEDIT 18 0 "Q4(z)" "-%#Q4G6#%\"zG" }{TEXT -1 0 "" }{TEXT 275 10 ", we find:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "QQ_zu:=(-u^2*z^2-2*u^2*z^3+u^4*z^6+u+1-u^3*z^4+u*z)* (-1+z)/(u*z^2-1)/(u^4*z^8-u^2*z^5-2*u^2*z^4-u*z^3-z+1);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&QQ_zuG**,0*&%\"uG\"\"#%\"zGF)!\"\"*&F(F)F*\" \"$!\"#*&F(\"\"%F*\"\"'\"\"\"F(F2F2F2*&F(F-F*F0F+*&F(F2F*F2F2F2,&F+F2F *F2F2,&*&F(F2F*F)F2F+F2F+,.*&F(F0F*\"\")F2*&F(F)F*\"\"&F+*&F(F)F*F0F.* &F(F2F*F-F+F*F+F2F2F+" }}}{PARA 0 "" 0 "" {TEXT -1 75 "We then automat ically obtained the main table in the solution published by " }{TEXT 267 11 "SIAM Review" }{TEXT -1 123 ". We can even detect errors: for i nstance, in the last line we must have 1261 (instead of 1211) and 4756 (instead of 4762)!" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "map(s eries,series(QQ_zu,z=0,21),u,infinity);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+O%\"zG+'%\"uG\"\"\"\"\"!F'\"\"\"F(+%F&F'\"\"\"\"\"\"F*\g"\"#F*\" \"$+)F&F'\"\"\"\"\"&\"\"#F'\"\"$\"\"%+)F&F'\"\"\"\"\"'\"\"#\"\"%\"\"$ \"\"&+)F&F'\"\"\"\"\"(\"\"#\"\"*\"\"$\"\"'+)F&F'\"\"\"\"\")\"\"#\"#;\" \"$\"\"(+-F&F'\"\"\"F@\"\"#\"#D\"\"$\"#:\"\"%F'\"\"&\"\")+-F&F'\"\"\" \"#5\"\"#\"#N\"\"$\"#S\"\"%FE\"\"&\"\"*+-F&F'\"\"\"\"#6\"\"#\"#Y\"\"$ \"#w\"\"%\"#J\"\"&\"#5+-F&F'\"\"\"\"#7\"\"#\"#e\"\"$\"$C\"\"\"%\"#&)\" \"&\"#6+1F&F'\"\"\"\"#8\"\"#\"#r\"\"$\"$&=\"\"%\"$!>\"\"&FW\"\"'F'\"\" (\"#7+1F&F'\"\"\"\"#9\"\"#Fjo\"\"$\"$g#\"\"%\"$g$\"\"&\"$a\"\"\"'F_p\" \"(\"#8+1F&F'\"\"\"FO\"\"#\"$+\"\"\"$\"$]$\"\"%\"$5'\"\"&\"$C%\"\"'F]o \"\"(\"#9+1F&F'\"\"\"FG\"\"#\"$;\"\"\"$\"$c%\"\"%\"$c*\"\"&\"$I*\"\"' \"$&H\"\"(\"#:+5F&F'\"\"\"\"#<\"\"#\"$L\"\"\"$\"$z&\"\"%\"%:9\"\"&\"%v <\"\"'\"$*))\"\"(\"#q\"\")F'\"\"*\"#;+5F&F'\"\"\"\"#=\"\"#\"$^\"\"\"$ \"$?(\"\"%\"%0?\"\"&\"%!3$\"\"'\"%)=#\"\"(\"$[%\"\")\"#>\"\"*\"#<+5F&F '\"\"\"Fdu\"\"#\"$q\"\"\"$\"$!))\"\"%\"%XF\"\"&\"%&)\\\"\"'\"%_Y\"\"( \"%p;\"\")\"$b\"\"\"*\"#=+5F&F'\"\"\"\"#?\"\"#Fep\"\"$\"%g5\"\"%\"%bO \"\"&\"%]w\"\"'\"%\"*))\h"\"(\"%=Z\"\")\"$0)\"\"*\"#>+9F&F'\"\"\"\"#@\" \"#\"$6#\"\"$\"%h7\"\"%\"%cZ\"\"&\"&c7\"\"\"'\"&'o:\"\"(\"&,7\"\"\")\" %OJ\"\"*\"$E\"\"#5F'\"#6\"#?+%F&-%\"OG6#F'F(\"#@" }}}{PARA 0 "" 0 "" {TEXT -1 72 "We have also easy access to any subproblem defined by fix ing the number " }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 0 "" }{TEXT -1 10 " of boxes:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "map(nor mal,series(QQ_zu,u=0,8));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+5%\"uG\" \"\"\"\"!,$*$,&!\"\"F%%\"zGF%F*F*\"\"\",$*(F+\"\"%,&!\"&F%F+F/F%F)!\"# F*\"\"#*(F+F/,(F*F%F+F**$F+\"\"&F%F%F)!\"$\"\"$*(F+\"\"),(\"#:F%F+!#?* $F+\"\"#\"\"'F%F)!\"%\"\"%,$*(F+F;,,F%F%F+\"\"$F?F%F6!#6*$F+FA\"\"(F%F )F1F*\"\"&*(F+\"#7,,\"#NF%F+!#cF?\"#D*$F+FGFBFIF%F%F)!\"'\"\"'*(F+FM,0 F*F%F+FSF?FSFRF*F6\"#mFI!#u*$F+FJ\"#@F%F)!\"(\"\"(-%\"OG6#F%\"\")" }}} }}}{MARK "2 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 } \"$q\"\"\"$\"$!))\"\"%\"%XF\"\"&\"%&)\\\"\"'\"%_Y\"\"( \"%p;\"\")\"$b\"\"\"*\"#=+5F&F'\"\"\"\"#?\"\"#Fep\"\"$\"%g5\"\"%\"%bO \"\"&\"%]w\"\"'\"%\"*))\hthem;9D!ky}xEtheor/+!ky}theorem']yE theoretic } theoretical+#'!kNjtherefor'+9Dy}x thermodynamicthes  +.!>?A9D+K]bce%ff!kmyy>}dW׵]xEj   thesitheta  theyc++.m6bfQi!kFsy׵xthickthicknesthinthing 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2 3 "DEC ALPHA UNIX" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "2 D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 " " 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYtLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "H eading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0u 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 60 "Enumerating alcohols an d other classes of chemical moleculs," }}{PARA 257 "" 0 "" {TEXT 258 26 "an example of Polya theory" }}{PARA 258 "" 0 "" {TEXT -1 47 "\nFre deric Chyzak\n(Version of January 13, 1997)\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "Alkanes are a simple class of chemical compounds. Th ey are generically described by the chemical formula " }{XPPEDIT 18 0 "C[n]*H[2*n+2]" "*&&%\"CG6#%\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F'F'\"\"#F' F'" }{TEXT -1 28 ". First examples for small " }{XPPEDIT 18 0 "n" "I \"nG6\"" }{TEXT -1 14 " are methane (" }{XPPEDIT 18 0 "n=1" "/%\"nG\" \"\"" }{TEXvT -1 11 "), ethane (" }{XPPEDIT 18 0 "n=2" "/%\"nG\"\"#" } {TEXT -1 12 "), propane (" }{XPPEDIT 18 0 "n=3" "/%\"nG\"\"$" }{TEXT -1 11 "), butane (" }{XPPEDIT 18 0 "n=4" "/%\"nG\"\"%" }{TEXT -1 23 ") , a.s.o. For a given " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 40 " h owever, there exist several different " }{TEXT 271 7 "isomers" }{TEXT -1 226 ", i.e., different structures of bonds between atoms. In chemi stry, there is much interest in knowing the number, or better yet the \+ list, of such isomers. Alcohols are obtained from alkanes by replacin g a hydrogen atom by an " }{XPPEDIT 18 0 "OH" "I#OHG6\"" }{TEXT -1 116 " group. It follows that they are isomorphic to carbon chains wit h a distinguished node, or again to alkyl radicals " }{XPPEDIT 18 0 "C [n]*H[2*n+1]" "*&&%\"CG6#%\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F'F'\"\"\"F'F '" }{TEXT -1 113 ", which are alkanes with a missing hydrogen atom. I f we disregard geometrical constraints (i.e., if we consider " }{TEXT 272 10 "structural" }{TEXT -1 23 " isomewrs only, and not " }{TEXT 273 14 "conformational" }{TEXT -1 152 " isomers), this leads to a pure gra ph-theoretical problem: how many rooted trees are there with n interna l nodes, where each internal node has degree 4?" }}{PARA 0 "" 0 "" {TEXT -1 35 "In this session, we thus consider " }{TEXT 257 6 "rooted " }{TEXT -1 68 " trees, so that we count and enumerate alkyls, with ge neric formula " }{XPPEDIT 18 0 "C[n]*H[2*n+1" "*&&%\"CG6#%\"nG\"\"\"&% \"HG6#,&*&\"\"#F'F&F'F'\"\"\"F'F'" }{TEXT -1 57 ". The combinatorics \+ also corresponds to simple alcohols " }{XPPEDIT 18 0 "C[n]*H[2*n+1]*O* H" "**&%\"CG6#%\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F'F'\"\"\"F'F'%\"OGF'F)F '" }{TEXT -1 27 ", organo-metalic compounds " }{XPPEDIT 18 0 "C[n]*H[2 *n+1]*X" "*(&%\"CG6#%\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F'F'\"\"\"F'F'%\"X GF'" }{TEXT -1 165 ", and any other monosubstituted alkanes. We next \+ treat the cases of disubstituted and trisubstituted alkanes. We devel op the study of our models using the package " }{HYPERLNK 17 "Comxbstru ct" 2 "combstruct" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "with(combstruct);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7.%+allstructsG%&countG%%drawG%)finishedG%'gfeqnsG%)gfseriesG%(gfsolve G%,iterstructsG%+nextstructG%,prog_gfeqnsG%.prog_gfseriesG%-prog_gfsol veG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "Enumerations of such clas ses of chemical compounds are part of Polya theory. We refer to the b ook by G. Polya and R. C. Read [" }{TEXT 266 67 "Combinatorial Enumera tion of Groups, Graphs, and Chemical Compounds" }{TEXT -1 54 ", (1987) , Springer-Verlag] for more extensive results." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "Monosubstituted alkanes, " }{XPPEDIT 18 0 "C[n]*H[2* n+1]*X" "*(&%\"CG6#%\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F'F'\"\"\"F'F'%\"XG F'" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "In this section, we study mo nosubstituted alkanes, i.e., " }{TEXT 275 6 "rooted" }{TEXT -1 60 " tr ees, first without any constraint, next according to the " }{TEXT 274 6 "height" }{TEXT -1 1 "y." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 "Gen eral alkyls" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Definition" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 280 "An alkyl radical can be viewed as a carbon atom linked to at most 3 alkyl radicals. Thus, we only take into account hydrogen atoms implicitly. There is no loss of informati on, since hydrogen atoms can always be recovered from the carbon skele ton. This yields the class equation " }{XPPEDIT 18 0 "Alkyl=Carbon*(E psilon+Alkyl+Alkyl^2+Alkyl^3)" "/%&AlkylG*&%'CarbonG\"\"\",*%(EpsilonG F&F#F&*$F#\"\"#F&*$F#\"\"$F&F&" }{TEXT -1 42 ", which we map into the \+ following grammar:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "gram m_Alkyl:=Alkyl=Prod(Carbon,Set(Alkyl,card<=3)),Carbon=Atom:\nspecs_Alk yl:=[Alkyl,\{gramm_Alkyl\},unlabelled]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Note that since the " }{HYPERLNK 17 "Set" 2 "combstruct[s pecification]" "" }{TEXT -1 190 " construct denotes multisets, i.e., s ets with repetitions, a carbon atom of an alkyl is allozwed to be bound to two copies of the same subtree (but the order of the subtrees does not matter)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "Define the size of an alkyl as the number of carbon atoms it contains. We compute th e number of alkyls of a given size using " }{HYPERLNK 17 "combstruct[c ount]" 2 "combstruct[count]" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "seq(count(specs_Alkyl,size=i),i=0..50);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6U\"\"!\"\"\"F$\"\"#\"\"%\"\")\"#<\"#R\" #*)\"$6#\"$2&\"%Q7\"%dI\"%Rw\"&T#>\"&l)[\"'1\\7\"')>@$\"'>-$)\"(5g:#\" (4@i&\")8er9\")_\"\\'Q\"*F>#=5\"*&[5!p#\"*nlc7(\"+WL*>*=\"+G[qM]\",1y6 DM\"\",p3bme$\",)Gl8*f*\"-1X'GLd#\"-0TNG4p\".f:N@y&=\".`rg0B+&\"/pc2Sk [8\"/y-V#Q/k$\"/$Gq\"z2Q)*\"0x/+_&ehE\"0Z9$o(z!3s\"1>=m]?+a>\"1jq,#p]> I&\"2<#R6h\"**)R9\"2_dY^(ox8R\"3l9`eY&fY1\"\"38%Rk5*QT)*G\"3hd4\\v6U'* y\"4cblrf%\\\"G:#\"4a>Lh$3eAte\"5u!RqbZ_\\Lg\"\"5gxdP>%\\b(zV" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "This series appears as the entry \+ " }{TEXT 261 5 {"M1146" }{TEXT -1 30 " (\"quartic planted trees with " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 56 " nodes\") in the book by N . J. A. Sloane and S. Plouffe [" }{TEXT 262 37 "The Encyclopedia of In teger Sequences" }{TEXT -1 26 ", (1995), Academic Press]." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Here is an example of an alkyl with 6 car bon atoms, obtained by the command " }{HYPERLNK 17 "combstruct[draw]" 2 "combstruct[draw]" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "alk:=draw(specs_Alkyl,size=6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$alkG-%%ProdG6$%'CarbonG-%$SetG6#-F&6$F(-F*6$-F&6$F(- F*6#-F&6$F(%(EpsilonGF0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "The fo llowing procedure rewrites an alkyl into a more readable way." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "nice:=proc(alk) eval(subs( \{Epsilon=NULL,Carbon=C,Prod=proc() global H; [args] end,Set=proc() ar gs end\},alk)) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nice (alk);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$%\"CG7%|F$7$F$7#F$F&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "The following procedure computes t he size of a given alkyl." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "size:=proc(alk) option remember; 1+convert(map(size,op(2,alk)),`+ `) end:\nsize(Prod(Carbon,Epsilon)):=1:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "The following procedure computes the height of a given al kyl." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "height:=proc(alk) \+ option remember; 1+max(op(map(height,op(2,alk)))) end:\nheight(Prod(Ca rbon,Epsilon)):=1:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Here is an \+ alkyl with 50 carbon atoms, its nice representation and height." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "alk:=draw(specs_Alkyl,size=5 0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nice(alk);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7$%\"CG7$F$7%F$7#F$7$F$7%F$7%F$7&F$F'7$F$7%F $F'7%F$F'F'7&F$7$F$7$F$F'F07%F$7&F$F'F'7&F$F'F'F17%F$F'F17$F$7$F$7%F$F 1F.7%F$F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "height(alk); " }}{PARA 1}1 "" 1 "" {XPPMATH 20 "6#\"#8" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Empirical study" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 7 "Dr awing" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 171 "By drawing several rando m structures, we can study probabilistic properties of alkyls. For in stance, the following is a probabilistic estimate of their height on a verage:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "for i to 10 do h o[i]:=height(draw(specs_Alkyl,size=50)) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#hoG6#\"\"\"\"#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%#hoG6#\"\"#\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#hoG6#\"\"$\" #>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#hoG6#\"\"%\"#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%#hoG6#\"\"&\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#hoG6#\"\"'\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#hoG6# \"\"(\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#hoG6#\"\")\"#;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#hoG6#\"\"*\"#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%#hoG6#\"#5\"#<" }}}{EXCHG {P~ARA 0 "> " 0 "" {MPLTEXT 1 0 23 "add(ho[i],i=1..10)/10.;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++I:!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "In the sam e way, we get a probabilistic estimate of their standard deviation:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sqrt(add((ho[i]-\")^2,i=1. .10)/10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+`%G=0#!\"*" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 22 "Exhaustive enumeration" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "The command " }{HYPERLNK 17 "combstruct[d raw]" 2 "combstruct[draw]" "" }{TEXT -1 20 " permits us to draw " } {TEXT 259 3 "one" }{TEXT -1 44 " structure at random. We can also gen erate " }{TEXT 260 3 "all" }{TEXT -1 31 " alkyls of a given size, usin g " }{HYPERLNK 17 "combstruct[allstructs]" 2 "combstruct[allstructs]" "" }{TEXT -1 222 ", so as to compute the mean of a particular paramete r exactly, or to count all those with a particular property. For inst ance, the height of trees cannot be represented in the class of combin atorial structures when using " }{HYPERLNK 17 "Combstruct" 2 "combstru ct" "" }{TEXT -1 134 ". For instance, by computing all alkyls of size 5, we get the distribution of height for these alkyls (in their nice \+ representation)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "allstru cts(specs_Alkyl,size=5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "map(nice,\");" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7*7$%\"CG7%F%7#F%7$F %F'7&F%F'F'F(7%F%F'7%F%F'F'7$F%7$F%7$F%F(7$F%7&F%F'F'F'7$F%7$F%F+7%F%F 'F.7%F%F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sort(map(hei ght,\"\"));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"$F$F$F$\"\"%F%F% \"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Here we count 4 alkyls \+ of size 5 and height 3, 3 alkyls of size 5 and height 4, and 1 alkyl o f size 5 and height 5." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "By the \+ same method, we get the exact mean and standard deviation of the heigh t for small sizes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "esd: =proc(n) local i,as,mean;\n as:=map(height,allstructs(specs_Alkyl,s ize=n));\n mean:=evalf(convert(as,`+`)/nops(as));\n nops(as),mea n,evalf(sqrt(add((i-mean)^2,i=as))/nops(as))\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "for i from 2 to 6 do i=esd(i) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"#6%\"\"\"$F$\"\"!F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/\"\"$6%\"\"#$\"+++++D!\"*$\"+1R`NN!#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/\"\"%6%F$$\"\"$\"\"!$\"+0R`NN!#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/\"\"&6%\"\")$\"++++DO!\"*$\"+huigC!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"'6%\"#<$\"+fqk " 0 "" {MPLTEXT 1 0 307 "gramm_ltd_height:=proc(n) option remembe r;\n Alkyl_height[n]=Prod(Carbon,Set(Alkyl_height[n-1],card<=3)),gr amm_ltd_height(n-1)\nend:\ngramm_ltd_height(1):=Alkyl_height[1]=Prod(C arbon,Epsilon),Carbon=Atom:\nspecs_ltd_height:=proc(n) option remember ;\n [Alkyl_height[n],\{gramm_ltd_height(n)\},unlabelled]\nend:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "The following procedure rewrites a n alkyl into a more readable way." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "nice:=proc(alk) eval(subs(\{Epsilon=NULL,Carbon=C,Pr od=proc() global H; [args] end,Set=proc() args end\},alk)) end:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "The following procedures compute t he size and height of a given alkyl." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "size:=proc(alk) option remember; 1+convert(map(size, op(2,alk)),`+`) end:\nsize(Prod(Carbon,Epsilon)):=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "height:=proc(alk) option remember; 1+max (op(map(height,op(2,alk)))) end:\nheight(Prod(Carbon,Epsilon)):=1:" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "For instance, we compute the heig ht of a random alkyl of size 10 and height at most 5." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "alk:=draw(specs_ltd_height(5),size= 10):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nice(alk);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7&%\"CG7#F$7$F$F%7$F$7&F$F%F%F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "size(alk),height(alk);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"#5\"\"&" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 213 "In this section, we proceed to compute a table of the nu mber of alkyls according to their size and height. The first method i s by generating all structures. Next, we use generating functions to \+ extend the table." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "Generating \+ all structures" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "The following pr ocedure remembers all the alkyls of a given size and with bounded heig ht." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "list_all_st:=proc(d, s) option remember; allstructs(specs_ltd_height(d),size=s) end:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "An alkyl of height " }{XPPEDIT 18 0 "h" "I\"hG6\"" }{TEXT -1 20 " has a size at most " }{XPPEDIT 18 0 "( 3^h-1)/2" "*&,&)\"\"$%\"hG\"\"\"\"\"\"!\"\"F'\"\"#F)" }{TEXT -1 56 ". \+ Therefore, to produce all alkyls with height at most " }{XPPEDIT 18 0 "h[max]" "&%\"hG6#%$maxG" }{TEXT -1 48 ", we need to produce all alk yls with size up to " }{XPPEDIT 18 0 "s[max]=(3^h[max]-1)/2" "/&%\"sG6 #%$maxG*&,&)\"\"$&%\"hG6#F&\"\"\"\"\"\"!\"\"F.\"\"#F0" }{TEXT -1 1 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "s[max]:=(3^h[max]-1)/2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"sG6#%$maxG,&)\"\"$&%\"hGF&#\" \"\"\"\"##!\"\"F/F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "To begin w ith, we enumerate all alkyls with height at most 3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "h[max]:=3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"hG6#%$maxG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "for i from 1 to s[max] do i,nops(list_all_st(h[max],i)),map(nice,l ist_all_st(h[max],i)) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\"F#7 #7#%\"CG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"#\"\"\"7#7$%\"CG7#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"$\"\"#7$7%%\"CG7#F'F(7$F'7$F'F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"%\"\"$7%7%%\"CG7#F'7$F'F(7$F'7 %F'F(F(7&F'F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"&\"\"%7&7&%\" CG7#F'F(7$F'F(7%F'F(7%F'F(F(7$F'7&F'F(F(F(7%F'F)F)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%\"\"'\"\"%7&7&%\"CG7#F'7$F'F(F)7%F'F(7&F'F(F(F(7%F'F) 7%F'F(F(7&F'F(F(F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6%\"\"(\"\"&7'7&% \"CG7$F'7#F'F(F(7&F'F)F)7&F'F)F)F)7&F'F)F(7%F'F)F)7%F'F-F-7%F'F(F+" }} {PARA 12 "" 1 "" {XPPMATH 20 "6%\"\")\"\"%7&7&%\"CG7#F'7%F'F(F(F)7%F'F )7&F'F(F(F(7&F'F(7$F'F(F+7&F'F-F-F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 %\"\"*\"\"%7&7&%\"CG7#F'7%F'F(F(7&F'F(F(F(7&F'7$F'F(F,F*7&F'F,F)F)7%F' F*F*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6%\"#5\"\"$7%7&%\"CG7$F'7#F'7%F' F)F)7&F'F)F)F)7&F'F)F+F+7&F'F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6% \"#6\"\"#7$7&%\"CG7$F'7#F'7&F'F)F)F)F*7&F'7%F'F)F)F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#7\"\"\"7#7&%\"CG7%F'7#F'F)7&F'F)F)F)F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%\"#8\"\"\"7#7&%\"CG7&F'7#F'F)F)F(F(" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "In this way, we have obtained t he truncation of the bivariate generating function of alkyls with size marked by " }{XPPEDIT 18 0 "z" "I\"zG6\"" }{TEXT -1 15 " and height b y " }{XPPEDIT 18 0 "u" "I\"uG6\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "enum_BGF:=map(series,series(convert(map(proc (s,z,u) z^size(s)*u^height(s) end,map(op,[seq(list_all_st(h[max],i),i= 1..s[max])]),z,u),`+`),z,infinity),u,infinity);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)enum_BGFG+=%\"zG+%%\"uG\"\"\"\"\"\"\"\"\"+%F(F)\"\"# \"\"#+'F(F)\"\"#F)\"\"$\"\"$+'F(F)\"\"#\"\"#\"\"$\"\"%+%F(\"\"%\"\"$\" \"&F8\"\"'+%F(\"\"&\"\"$\"\"(F8\"\")F8\"\"*+%F(\"\"$\"\"$\"#5+%F(F5\" \"$\"#6+%F(F)\"\"$\"#7FJ\"#8" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Generating functions" }}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "combstru ct[gfeqns]" 2 "combstruct[gfeqns]" "" }{TEXT -1 210 " returns a system of functional equations satisfied by the generating functions of rela ted combinatorial structures. In the case of the alkyls with maximum \+ height above, we get the following triangular system." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "gfeqns(op(2..3,specs_ltd_height(4)) ,z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7'/-%'CarbonG6#%\"zGF(/-&%-Alk yl_heightG6#\"\"#F'*&F%\"\"\",0F0F0-&F,6#F0F'F0-F36#*$F(F.#F0F.*$F2F.F 8-F36#*$F(\"\"$#F0F=*&F2F0F5F0F8*$F2F=#F0\"\"'F0/-&F,6#F=F'*&F%F0,0F0F 0F*F0-F+F6F8*$F*F.F8-F+F;F>*&F*F0FIF0F8*$F*F=FAF0/F2F%/-&F,6#\"\"%F'*& F%F0,0F0F0FDF0-FEF6F8*$FDF.F8-FEF;F>*&FDF0FVF0F8*$FDF=FAF0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "gfsol:=gfsolve(op(2..3,specs_ltd_he ight(4)),z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&gfsolG<'/-&%-Alkyl_ heightG6#\"\"$6#%\"zG,\"$P\"F-F/F8\"$%=F6\" #qF<\"#**FFF1F0\"#?*$F-\"#:\"$+$*$F-\"#=\"$)\\*$F-FR\"$%f*$F-\"#E\"$`% *$F-\"#C\"$q&*$F-\"#;\"$p$*$F-\"#@\"$9'*$F-\"#F\"$y$*$F-\"#I\"$\"=*$F- \"#RF/*$F-\"#L\"#c*$F-\"#OF?*$F-\"#A\"$C'*$F-\"#9\"$R#*$F-\"#QF+*$F-\" #PFA*$F-\"#MFgp*$F-\"#NFR*$F-FL\"$G\"*$F-\"#G\"$7$*$F-\"#H\"$Q#*$F-\"# K\"#*)*$F-\"#B\"$,'*$F-\"#D\"$9&*$F-\"#<\"$K%*$F-\"#>\"$^&*$F-\"#SF/F: F/FDF;FEF5/-&F)6#F/F,F-/-%'CarbonGF,F-/-&F)6#F;F,,*F-F/F:F/FDF/FEF/" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 151 "In particular, we have obtained a truncation of the bivariate generating function of all alkyls (i.e. , with no constraint on height). In this series, " }{XPPEDIT 18 0 "u " "I\"uG6\"" }{TEXT -1 55 " marks the height. It extends the previous truncation " }{XPPEDIT 18 0 "enum_BGF" "I)enum_BGFG6\"" }{TEXT -1 1 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "BGF:=map(series,serie s(eval(subs(Alkyl_height[0]=0,gfsol,add(u^h*(Alkyl_height[h]-Alkyl_hei ght[h-1])(z),h=1..4))),z,infinity),u,infinity);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$BGFG+]p%\"zG+%%\"uG\"\"\"\"\"\"\"\"\"+%F(F)\"\"#\"\" #+'F(F)\"\"#F)\"\"$\"\"$+)F(F)\"\"#\"\"#\"\"$F)\"\"%\"\"%+'F(\"\"%\"\" $\"\"$\"\"%\"\"&+'F(F:\"\"$\"\")\"\"%\"\"'+'F(\"\"&\"\"$\"#:\"\"%\"\"( +'F(F:\"\"$\"#F\"\"%\"\")+'F(F:\"\"$\"#V\"\"%\"\"*+'F(F<\"\"$\"#n\"\"% \"#5+'F(F5\"\"$\"#(*\"\"%\"#6+'F(F)\"\"$\"$O\"\"\"%\"#7+'F(F)\"\"$\"$$ =\"\"%\"#8+%F(\"$R#\"\"%\"#9+%F(\"$+$\"\"%\"#:+%F(\"$p$\"\"%\"#;+%F(\" $K%\"\"%\"#<+%F(\"$)\\\"\"%\"#=+%F(\"$^&\"\"%\"#>+%F(\"$%f\"\"%\"#?+%F (\"$9'\"\"%\"#@+%F(\"$C'\"\"%\"#A+%F(\"$,'\"\"%\"#B+%F(\"$q&\"\"%\"#C+ %F(\"$9&\"\"%\"#D+%F(\"$`%\"\"%\"#E+%F(\"$y$\"\"%\"#F+%F(\"$7$\"\"%\"# G+%F(\"$Q#\"\"%\"#H+%F(\"$\"=\"\"%\"#I+%F(\"$G\"\"\"%\"#J+%F(\"#*)\"\" %\"#K+%F(\"#c\"\"%\"#L+%F(\"#P\"\"%\"#M+%F(\"#?\"\"%\"#N+%F(\"#7\"\"% \"#O+%F(\"\"'\"\"%\"#P+%F(F<\"\"%\"#Q+%F(F)\"\"%\"#RFeu\"#S" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "This is made explicit on the follo wing normalized difference: each entry starts with a term in " } {XPPEDIT 18 0 "u^4" "*$%\"uG\"\"%" }{TEXT -1 41 ", denoting alkyls wit h height at least 4." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "map (series,series(BGF-enum_BGF,z,infinity),u,infinity);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+go%\"zG+%%\"uG\"\"\"\"\"%\"\"%+%F&\"\"$\"\"%\"\"&+% F&\"\")\"\"%\"\"'+%F&\"#:\"\"%\"\"(+%F&\"#F\"\"%\"\")+%F&\"#V\"\"%\"\" *+%F&\"#n\"\"%\"#5+%F&\"#(*\"\"%\"#6+%F&\"$O\"\"\"%\"#7+%F&\"$$=\"\"% \"#8+%F&\"$R#\"\"%\"#9+%F&\"$+$\"\"%\"#:+%F&\"$p$\"\"%\"#;+%F&\"$K%\" \"%\"#<+%F&\"$)\\\"\"%\"#=+%F&\"$^&\"\"%\"#>+%F&\"$%f\"\"%\"#?+%F&\"$9 '\"\"%\"#@+%F&\"$C'\"\"%\"#A+%F&\"$,'\"\"%\"#B+%F&\"$q&\"\"%\"#C+%F&\" $9&\"\"%\"#D+%F&\"$`%\"\"%\"#E+%F&\"$y$\"\"%\"#F+%F&\"$7$\"\"%\"#G+%F& \"$Q#\"\"%\"#H+%F&\"$\"=\"\"%\"#I+%F&\"$G\"\"\"%\"#J+%F&\"#*)\"\"%\"#K +%F&\"#c\"\"%\"#L+%F&\"#P\"\"%\"#M+%F&\"#?\"\"%\"#N+%F&\"#7\"\"%\"#O+% F&\"\"'\"\"%\"#PF*\"#QF%\"#RF%\"#S" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 "Table of the number of alkyls according to size and height" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 235 "Calculations with respect to diff erent heights are much more efficient than the method of exhaustive en umeration. This makes it possible for us to set up the table of the n umber of alkyls according to size and height in a few minutes:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "h[max]:=5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"hG6#%$maxG\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "gfsol:=gfsolve(op(2..3,specs_ltd_height(h[max])),z); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&gfsolG<(/-&%-Alkyl_heightG6#\" \"&6#%\"zG,^z*$F-\"\"'\"#;*$F-\"\")\"#j*$F-\"\"*\"$@\"*$F-\"#7\"$\\(F- \"\"\"*$F-\"#8\"%W8*$F-\"#5\"$D#*$F-\"#6\"$:%*$F-F+F3*$F-\"\"(\"#L*$F- \"#$*\")yF#*R*$F-\"#%)\"*Nl/_&*$F-\"#()\"*3Mrh#*$F-\"#'*\")kzn7*$F-\"# :\"%HT*$F-\"#=\"&a.#*$F-\"#?\"&1d&*$F-\"#E\"'FF()*$F-\"#C\"'bXO*$F-F1 \"%1r*$F-\"#@\"&G1**$F-\"#F\"(X&G8*$F-\"#I\"((G$R%*$F-\"#R\")\\av*)*$F -FH\")EX?8*$F-\"#O\")-^4O*$F-\"#A\"'Hd9*$F-\"#9\"%lB*$F-\"#Q\")^9&p'*$ F-\"#P\"))**=%\\*$F-\"#M\")0zl=*$F-\"#N\")W#)3E*$F-\"#J\"($o2k*$F-\"#G \"(O0+#*$F-\"#H\"(a0)H*$F-\"#K\"(IoC**$F-\"#B\"',=B*$F-\"#D\"'1sc*$F- \"#<\"&/@\"*$F-\"#>\"&$)Q$*$F-\"#S\"*\\J1>\"*$F-\"\"#F;*$F-\"\"$F_s*$F -F7F;*$F-\"#p\"+'enS*R*$F-\"#v\"+YN@KD*$F-\"#^\"+QR_B8*$F-\"#d\"+rF*)[ G*$F-\"$?\"F;*$F-\"#i\"+XuV$*R*$F-\"#c\"+b:CwD*$F-\"#e\"+R[0;J*$F-\"#k \"+7L\"yA%*$F-\"#Y\"*\"Qf&>&*$F-\"#]\"+1q`@6*$F-\"#!)\"+*$ F-\"#*)\"*k_J[\"*$F-\"#\"*\")/!)Hz*$F-\"##*\")'>&pc*$F-\"#%*\")n`nF*$F -\"#&*\"):]))=*$F-\"#(*\"(+GP)*$F-\"#)*\"(EcV&*$F-\"$+\"\"(&Ru@*$F-\"$ ,\"\"(!zQ8*$F-\"$.\"\"'R$z%*$F-\"$/\"\"'!Gy#*$F-\"$1\"\"&/#))*$F-\"$2 \"\"&E\"[*$F-\"$4\"\"&[L\"*$F-\"$5\"\"%Wn*$F-\"$7\"\"%/;*$F-\"$8\"\"$e (*$F-\"$:\"\"$a\"*$F-\"$;\"Fbt*$F-\"$=\"F@*$F-\"$>\"F]z*$F-\"#f\"+T8+r L*$F-\"#h\"+7KQ;Q*$F-\"#l\"+R<(pF%*$F-\"#n\"+;rgHU*$F-\"#r\"+d_:*f$*$F -\"#t\"+b3u#4$*$F-\"#x\"+#HbP(>*$F-\"#z\"+wa@j9*$F-\"#T\"*IZGc\"*$F-\" #V\"*ee'3E*$F-\"#Z\"*<&R*R'*$F-\"#\\\"*_nBS**$F-\"#`\"+j!*e%y\"*$F-\"# b\"+N2S/B/-&F)6#FasF,, " 0 "" {MPLTEXT 1 0 147 "BGF:=map (series,series(eval(subs(Alkyl_height[0]=0,gfsol,add(u^hh*(Alkyl_heigh t[hh]-Alkyl_height[hh-1])(z),hh=1..h[max]))),z,infinity),u,infinity); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$BGFG+_z%\"zG+%%\"uG\"\"\"\"\"\" \"\"\"+%F(F)\"\"#\"\"#+'F(F)\"\"#F)\"\"$\"\"$+)F(F)\"\"#\"\"#\"\"$F)\" \"%\"\"%+)F(\"\"%\"\"$\"\"$\"\"%F)\"\"&\"\"&+)F(F:\"\"$\"\")\"\"%F:\" \"&\"\"'+)F(\"\"&\"\"$\"#:\"\"%\"#8\"\"&\"\"(+)F(F:\"\"$\"#F\"\"%\"#K \"\"&\"\")+)F(F:\"\"$\"#V\"\"%\"#u\"\"&\"\"*+)F(F<\"\"$\"#n\"\"%\"$b\" \"\"&\"#5+)F(F5\"\"$\"#(*\"\"%\"$;$\"\"&\"#6+)F(F)\"\"$\"$O\"\"\"%\"$7 '\"\"&\"#7+)F(F)\"\"$\"$$=\"\"%\"%g6\"\"&\"#8+'F(\"$R#\"\"%\"%E@\"\"& \"#9+'F(\"$+$\"\"%\"%HQ\"\"&\"#:+'F(\"$p$\"\"%\"%Pn\"\"&\"#;+'F(\"$K% \"\"%\"&s;\"\"\"&\"#<+'F(\"$)\\\"\"%\"&c)>\"\"&\"#=+'F(\"$^&\"\"%\"&KL $\"\"&\"#>+'F(\"$%f\"\"%\"&7^&\"\"&\"#?+'F(\"$9'\"\"%\"&9+*\"\"&\"#@+' F(\"$C'\"\"%\"'0^9\"\"&\"#A+'F(\"$,'\"\"%\"'+7B\"\"&\"#B+'F(\"$q&\"\"% \"'&)RO\"\"&\"#C+'F(\"$9&\"\"%\"'#pm&\"\"&\"#D+'F(\"$`%\"\"%\"'uA()\" \"&\"#E+'F(\"$y$\"\"%\"(n\"G8\"\"&\"#F+'F(\"$7$\"\"%\"(C-+#\"\"&\"#G+' F(\"$Q#\"\"%\"(;.)H\"\"&\"#H+'F(\"$\"=\"\"%\"(1JR%\"\"&\"#I+'F(\"$G\" \"\"%\"(bvS'\"\"&\"#J+'F(\"#*)\"\"%\"(TnC*\"\"&\"#K+'F(\"#c\"\"%\")qW? 8\"\"&\"#L+'F(\"#P\"\"%\")oyl=\"\"&\"#M+'F(\"#?\"\"%\")C#)3E\"\"&\"#N+ 'F(\"#7\"\"%\")!4&4O\"\"&\"#O+'F(\"\"'\"\"%\")#**=%\\\"\"&\"#P+'F(F<\" \"%\")[9&p'\"\"&\"#Q+'F(F)\"\"%\")[av*)\"\"&\"#R+'F(F)\"\"%\"*[J1>\"\" \"&\"#S+%F(\"*IZGc\"\"\"&\"#T+%F(\"*US)H?\"\"&\"#U+%F(\"*ee'3E\"\"&\"# V+%F(\"*VerJ$\"\"&\"#W+%F(\"*-)ftT\"\"&\"#X+%F(\"*\"Qf&>&\"\"&\"#Y+%F( \"*<&R*R'\"\"&\"#Z+%F(\"*HI%)z(\"\"&\"#[+%F(\"*_nBS*\"\"&\"#\\+%F(\"+1 q`@6\"\"&\"#]+%F(\"+QR_B8\"\"&\"#^+%F(\"+[z;X:\"\"&\"#_+%F(\"+j!*e%y\" \"\"&\"#`+%F(\"+z*G*Q?\"\"&\"#a+%F(\"+N2S/B\"\"&\"#b+%F(\"+b:CwD\"\"& \"#c+%F(\"+rF*)[G\"\"&\"#d+%F(\"+R[0;J\"\"&\"#e+%F(\"+T8+rL\"\"&\"#f+% F(\"+L/t1O\"\"&\"#g+%F(\"+7KQ;Q\"\"&\"#h+%F(\"+XuV$*R\"\"&\"#i+%F(\"+h 'p@8%\"\"&\"#j+%F(\"+7L\"yA%\"\"&\"#k+%F(\"+R<(pF%\"\"&\"#l+%F(\"+?Pmx U\"\"&\"#m+%F(\"+;rgHU\"\"&\"#n+%F(\"+.h3MT\"\"&\"#o+%F(\"+'enS*R\"\"& \"#p+%F(\"+[\"yQ\"Q\"\"&\"#q+%F(\"+d_:*f$\"\"&\"#r+%F(\"+7RQcL\"\"&\"# s+%F(\"+b3u#4$\"\"&\"#t+%F(\"+7,d:G\"\"&\"#u+%F(\"+YN@KD\"\"&\"#v+%F( \"+uI_\\A\"\"&\"#w+%F(\"+#HbP(>\"\"&\"#x+%F(\"+#)\\?5<\"\"&\"#y+%F(\"+ wa@j9\"\"&\"#z+%F(\"+\" \"&\"#))+%F(\"*k_J[\"\"\"&\"#*)+%F(\"*%R]#4\"\"\"&\"#!*+%F(\")/!)Hz\" \"&\"#\"*+%F(\")'>&pc\"\"&\"##*+%F(\")yF#*R\"\"&\"#$*+%F(\")n`nF\"\"& \"#%*+%F(\"):]))=\"\"&\"#&*+%F(\")kzn7\"\"&\"#'*+%F(\"(+GP)\"\"&\"#(*+ %F(\"(EcV&\"\"&\"#)*+%F(\"(N\"pM\"\"&\"#**+%F(\"(&Ru@\"\"&\"$+\"+%F(\" (!zQ8\"\"&\"$,\"+%F(\"'Z&3)\"\"&\"$-\"+%F(\"'R$z%\"\"&\"$.\"+%F(\"'!Gy #\"\"&\"$/\"+%F(\"'w%e\"\"\"&\"$0\"+%F(\"&/#))\"\"&\"$1\"+%F(\"&E\"[\" \"&\"$2\"+%F(\"&%eD\"\"&\"$3\"+%F(\"&[L\"\"\"&\"$4\"+%F(\"%Wn\"\"&\"$5 \"+%F(\"%`L\"\"&\"$6\"+%F(\"%/;\"\"&\"$7\"+%F(\"$e(\"\"&\"$8\"+%F(\"$Q $\"\"&\"$9\"+%F(\"$a\"\"\"&\"$:\"+%F(\"#i\"\"&\"$;\"+%F(FP\"\"&\"$<\"+ %F(\"#5\"\"&\"$=\"+%F(F:\"\"&\"$>\"+%F(F)\"\"&\"$?\"F[^m\"$@\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "In the following table, the entry \+ at row " }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 12 " and column " } {XPPEDIT 18 0 "c" "I\"cG6\"" }{TEXT -1 33 " is the number of alkyls of size " }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 0 "" }{TEXT -1 12 " an d height " }{XPPEDIT 18 0 "c" "I\"cG6\"" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "matrix([[` `,seq(`height = `.hh,hh =1..h[max])],seq([`size = `.ss,seq(coeff(coeff(BGF,z,ss),u,hh),hh=1..h [max])],ss=1..s[max])]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG 6#7fr7(%\"~G%+height~=~1G%+height~=~2G%+height~=~3G%+height~=~4G%+heig ht~=~5G7(%)size~=~1G\"\"\"\"\"!F1F1F17(%)size~=~2GF1F0F1F1F17(%)size~= ~3GF1F0F0F1F17(%)size~=~4GF1F0\"\"#F0F17(%)size~=~5GF1F1\"\"%\"\"$F07( %)size~=~6GF1F1F;\"\")F;7(%)size~=~7GF1F1\"\"&\"#:\"#87(%)size~=~8GF1F 1F;\"#F\"#K7(%)size~=~9GF1F1F;\"#V\"#u7(%*size~=~10GF1F1F<\"#n\"$b\"7( %*size~=~11GF1F1F8\"#(*\"$;$7(%*size~=~12GF1F1F0\"$O\"\"$7'7(%*size~=~ 13GF1F1F0\"$$=\"%g67(%*size~=~14GF1F1F1\"$R#\"%E@7(%*size~=~15GF1F1F1 \"$+$\"%HQ7(%*size~=~16GF1F1F1\"$p$\"%Pn7(%*size~=~17GF1F1F1\"$K%\"&s; \"7(%*size~=~18GF1F1F1\"$)\\\"&c)>7(%*size~=~19GF1F1F1\"$^&\"&KL$7(%*s ize~=~20GF1F1F1\"$%f\"&7^&7(%*size~=~21GF1F1F1\"$9'\"&9+*7(%*size~=~22 GF1F1F1\"$C'\"'0^97(%*size~=~23GF1F1F1\"$,'\"'+7B7(%*size~=~24GF1F1F1 \"$q&\"'&)RO7(%*size~=~25GF1F1F1\"$9&\"'#pm&7(%*size~=~26GF1F1F1\"$`% \"'uA()7(%*size~=~27GF1F1F1\"$y$\"(n\"G87(%*size~=~28GF1F1F1\"$7$\"(C- +#7(%*size~=~29GF1F1F1\"$Q#\"(;.)H7(%*size~=~30GF1F1F1\"$\"=\"(1JR%7(% *size~=~31GF1F1F1\"$G\"\"(bvS'7(%*size~=~32GF1F1F1\"#*)\"(TnC*7(%*size ~=~33GF1F1F1\"#c\")qW?87(%*size~=~34GF1F1F1\"#P\")oyl=7(%*size~=~35GF1 F1F1\"#?\")C#)3E7(%*size~=~36GF1F1F1\"#7\")!4&4O7(%*size~=~37GF1F1F1\" \"'\")#**=%\\7(%*size~=~38GF1F1F1F<\")[9&p'7(%*size~=~39GF1F1F1F0\")[a v*)7(%*size~=~40GF1F1F1F0\"*[J1>\"7(%*size~=~41GF1F1F1F1\"*IZGc\"7(%*s ize~=~42GF1F1F1F1\"*US)H?7(%*size~=~43GF1F1F1F1\"*ee'3E7(%*size~=~44GF 1F1F1F1\"*VerJ$7(%*size~=~45GF1F1F1F1\"*-)ftT7(%*size~=~46GF1F1F1F1\"* \"Qf&>&7(%*size~=~47GF1F1F1F1\"*<&R*R'7(%*size~=~48GF1F1F1F1\"*HI%)z(7 (%*size~=~49GF1F1F1F1\"*_nBS*7(%*size~=~50GF1F1F1F1\"+1q`@67(%*size~=~ 51GF1F1F1F1\"+QR_B87(%*size~=~52GF1F1F1F1\"+[z;X:7(%*size~=~53GF1F1F1F 1\"+j!*e%y\"7(%*size~=~54GF1F1F1F1\"+z*G*Q?7(%*size~=~55GF1F1F1F1\"+N2 S/B7(%*size~=~56GF1F1F1F1\"+b:CwD7(%*size~=~57GF1F1F1F1\"+rF*)[G7(%*si ze~=~58GF1F1F1F1\"+R[0;J7(%*size~=~59GF1F1F1F1\"+T8+rL7(%*size~=~60GF1 F1F1F1\"+L/t1O7(%*size~=~61GF1F1F1F1\"+7KQ;Q7(%*size~=~62GF1F1F1F1\"+X uV$*R7(%*size~=~63GF1F1F1F1\"+h'p@8%7(%*size~=~64GF1F1F1F1\"+7L\"yA%7( %*size~=~65GF1F1F1F1\"+R<(pF%7(%*size~=~66GF1F1F1F1\"+?PmxU7(%*size~=~ 67GF1F1F1F1\"+;rgHU7(%*size~=~68GF1F1F1F1\"+.h3MT7(%*size~=~69GF1F1F1F 1\"+'enS*R7(%*size~=~70GF1F1F1F1\"+[\"yQ\"Q7(%*size~=~71GF1F1F1F1\"+d_ :*f$7(%*size~=~72GF1F1F1F1\"+7RQcL7(%*size~=~73GF1F1F1F1\"+b3u#4$7(%*s ize~=~74GF1F1F1F1\"+7,d:G7(%*size~=~75GF1F1F1F1\"+YN@KD7(%*size~=~76GF 1F1F1F1\"+uI_\\A7(%*size~=~77GF1F1F1F1\"+#HbP(>7(%*size~=~78GF1F1F1F1 \"+#)\\?5<7(%*size~=~79GF1F1F1F1\"+wa@j97(%*size~=~80GF1F1F1F1\"+7(%*size~=~89GF1F1F1F1\"*k_J[\"7(%*s ize~=~90GF1F1F1F1\"*%R]#4\"7(%*size~=~91GF1F1F1F1\")/!)Hz7(%*size~=~92 GF1F1F1F1\")'>&pc7(%*size~=~93GF1F1F1F1\")yF#*R7(%*size~=~94GF1F1F1F1 \")n`nF7(%*size~=~95GF1F1F1F1\"):]))=7(%*size~=~96GF1F1F1F1\")kzn77(%* size~=~97GF1F1F1F1\"(+GP)7(%*size~=~98GF1F1F1F1\"(EcV&7(%*size~=~99GF1 F1F1F1\"(N\"pM7(%+size~=~100GF1F1F1F1\"(&Ru@7(%+size~=~101GF1F1F1F1\"( !zQ87(%+size~=~102GF1F1F1F1\"'Z&3)7(%+size~=~103GF1F1F1F1\"'R$z%7(%+si ze~=~104GF1F1F1F1\"'!Gy#7(%+size~=~105GF1F1F1F1\"'w%e\"7(%+size~=~106G F1F1F1F1\"&/#))7(%+size~=~107GF1F1F1F1\"&E\"[7(%+size~=~108GF1F1F1F1\" &%eD7(%+size~=~109GF1F1F1F1\"&[L\"7(%+size~=~110GF1F1F1F1\"%Wn7(%+size ~=~111GF1F1F1F1\"%`L7(%+size~=~112GF1F1F1F1\"%/;7(%+size~=~113GF1F1F1F 1\"$e(7(%+size~=~114GF1F1F1F1\"$Q$7(%+size~=~115GF1F1F1F1\"$a\"7(%+siz e~=~116GF1F1F1F1\"#i7(%+size~=~117GF1F1F1F1FG7(%+size~=~118GF1F1F1F1\" #57(%+size~=~119GF1F1F1F1F;7(%+size~=~120GF1F1F1F1F07(%+size~=~121GF1F 1F1F1F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "A (huge) table for " } {XPPEDIT 18 0 "h[max]=7" "/&%\"hG6#%$maxG\"\"(" }{TEXT -1 43 " could b e computed in less than 10 minutes." }}}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "Disubstituted alkanes, " }{XPPEDIT 18 0 "C[n]*H[2*n]*X*Y " "**&%\"CG6#%\"nG\"\"\"&%\"HG6#*&\"\"#F'F&F'F'%\"XGF'%\"YGF'" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Enumerating disubstituted alkanes \+ " }{XPPEDIT 18 0 "C[n]*H[2*n]*X*Y" "**&%\"CG6#%\"nG\"\"\"&%\"HG6#*&\" \"#F'F&F'F'%\"XGF'%\"YGF'" }{TEXT -1 53 " is equivalent to enumerating monosubstituted alkyls " }{XPPEDIT 18 0 "C[n]*H[2*n]*X" "*(&%\"CG6#% \"nG\"\"\"&%\"HG6#*&\"\"#F'F&F'F'%\"XGF'" }{TEXT -1 163 ". The latter can generically be viewed as a carbon atom linked to one monosubstitu ted alkyl and at least 2 nonsubstituted alkyls. This yields the class equation " }{XPPEDIT 18 0 "S1_Alkyl[X]=Carbon*S1_Alkyl[X]*(Epsilon+Al kyl)+Carbon*X*(Epsilon+Alkyl+Alkyl^2)" "/&%)S1_AlkylG6#%\"XG,&*(%'Carb onG\"\"\"&F$6#F&F*,&%(EpsilonGF*%&AlkylGF*F*F**(F)F*F&F*,(F.F*F/F**$F/ \"\"#F*F*F*" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "gramm_S1_Alkyl:=S1_Alkyl[X]=Union(Prod(Carbon,S1_Alkyl[X],Set(Alk yl,card<=2)),Prod(Prod(Carbon,X),Set(Alkyl,card<=2))),X=Epsilon:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "specs_S1_Alkyl:=[S1_Alkyl[X] ,\{gramm_S1_Alkyl,gramm_Alkyl\},unlabelled]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "seq(count(specs_S1_Alkyl,size=i),i=0..50);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6U\"\"!\"\"\"\"\"#\"\"&\"#7\"#J\"#!)\"$5 #\"$b&\"%z9\"%fR\"&_1\"\"&g(G\"&5z(\"'C;@\"'@id\"(5Ad\"\"(LxH%\")Gtw6 \"),oEK\")EYf))\"*>\\aV#\"*B'G-n\"+PRGY=\"+=^g!4&\",o!ow/9\",$HA\\zQ\" -d!Q_@2\"\"-/(y9]'H\"-&HhiZ?)\".j#eksrA\".c\"HIL$H'\"/;M;oJW<\"/amji+P [\"0yf)4p!>M\"\"0y:)Q$eVs$\"1%R3p')yS.\"\"1\\4P:<\"3/&4SfWxAx%\"4Za4)pc)[!G8\"4Ir?\\oJH np$\"5D\\EIwK[FH5\"5h^#)=Pi&*[mG\"5,<39N`J\"\\)z\"6Pfmzfs!yxCA\"6N[u% \\T_!e+?'\"7n'yl!f+)G*>G<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "This series appears as the entry " }{TEXT 264 5 "M1418" }{TEXT -1 18 " (\" paraffins with " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 63 " carbon a toms\") in the book by N. J. A. Sloane and S. Plouffe [" }{TEXT 265 37 "The Encyclopedia of Integer Sequences" }{TEXT -1 26 ", (1995), Aca demic Press]." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 161 "Of course, ther e are more monosubstituted alkyls than nonsubstituted ones. We give t he ratios number of monosubstituted alkyls/number of alkyls for small \+ sizes:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "seq([i=evalf(coun t(specs_S1_Alkyl,size=i)/count(specs_Alkyl,size=i))],i=1..50);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6T7#/\"\"\"$F%\"\"!7#/\"\"#$F*F'7#/\"\"$ $\"+++++D!\"*7#/\"\"%$F.F'7#/\"\"&$\"++++vQF17#/\"\"'$\"+`B)eq%F17#/\" \"($\"+&Q:YQ&F17#/\"\")$\"+c]&fB'F17#/\"\"*$\"+t'y%4qF17#/\"#5$\"+,&y' 3yF17#/\"#6$\"+B.?/')F17#/\"#7$\"+ei\"zS*F17#/\"#8$\"+#*y*)>5!\")7#/\" #9$\"+v'f)*4\"F]o7#/\"#:$\"+p+@z6F]o7#/\"#;$\"+`Xre7F]o7#/\"#<$\"+/B.Q 8F]o7#/\"#=$\"+zjP<9F]o7#/\"#>$\"+Hzf'\\\"F]o7#/\"#?$\"+ya#ed\"F]o7#/ \"#@$\"+iy)\\l\"F]o7#/\"#A$\"+\"RNTt\"F]o7#/\"#B$\"+%)yC8=F]o7#/\"#C$ \"++XM#*=F]o7#/\"#D$\"+^$=9(>F]o7#/\"#E$\"+$zy/0#F]o7#/\"#F$\"+5Q_H@F] o7#/\"#G$\"+Uxb3AF]o7#/\"#H$\"+01e(G#F]o7#/\"#I$\"+\"f%fmBF]o7#/\"#J$ \"+#>+cW#F]o7#/\"#K$\"+)o)fCDF]o7#/\"#L$\"+I1f.EF]o7#/\"#M$\"+Eod#o#F] o7#/\"#N$\"+rxbhFF]o7#/\"#O$\"+=S`SGF]o7#/\"#P$\"+vf]>HF]o7#/\"#Q$\"+[ SZ)*HF]o7#/\"#R$\"+l&Qu2$F]o7#/\"#S$\"+K)*RcJF]o7#/\"#T$\"+9\"e`B$F]o7 #/\"#U$\"+^OJ9LF]o7#/\"#V$\"+bmE$R$F]o7#/\"#W$\"+>t@sMF]o7#/\"#X$\"+:e ;^NF]o7#/\"#Y$\"+*H7,j$F]o7#/\"#Z$\"+6p04PF]o7#/\"#[$\"+!y**zy$F]o7#/ \"#\\$\"+C5%p'QF]o7#/\"#]$\"+Z2)e%RF]o" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Here is an example of a monosubstituted alkyl with 6 carb on atoms, obtained by the command " }{HYPERLNK 17 "combstruct[draw]" 2 "combstruct[draw]" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "alk:=draw(specs_S1_Alkyl,size=6);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$alkG-%%ProdG6%%'CarbonG-F&6%F(-F&6$-F&6$F(%\"XG-%$ SetG6#-F&6$F(-F16#-F&6$F(%(EpsilonGF5F9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "The following procedure rewrites a monosubstituted alkyl \+ into a more readable way." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "nice:=proc(alk) subs([C,X]=CX,eval(subs(\{Epsilon=NULL,Carbon=C,P rod=proc() [args] end,Set=proc() args end\},alk))) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nice(alk);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$%\"CG7%F$7$%#CXG7$F$7#F$F)" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 86 "The following procedures compute the size and height of a given monosubstituted alkyl." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "size:=proc(alk) option remember; 1+convert(map(op,map2(map,si ze,[op(2..-1,alk)])),`+`) end:\nsize(Carbon):=1:\nsize(X):=0:\nsize(Ep silon):=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "height:=proc (alk) option remember; `if`(nops(alk)=2,1+max(op(map(height,op(2,alk)) )),1+max(height(op(2,alk)),op(map(height,op(3,alk))))) end:\nheight(Ca rbon):=1:\nheight(X):=0:\nheight(Epsilon):=0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Here is a monosubstituted alkyl with 50 carbon atoms, \+ its nice representation and height." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "alk:=draw(specs_S1_Alkyl,size=50):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nice(alk);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7&%\"CG7%F$7$F$7$F$7&F$7#%#CXG7#F$7%F$F+7$F$7%F$7$F$7$F$F+7%F$F0 F0F+7%F$F+7&F$F+F+F07$F$7&F$F+F37%F$7$F$F/7$F$7&F$F07%F$F+F+F:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "height(alk);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#6" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Tri substituted alkanes, " }{XPPEDIT 18 0 "C[n]*H[2*n-1]*X*Y*Z" "*,&%\"CG6 #%\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F'F'\"\"\"!\"\"F'%\"XGF'%\"YGF'%\"ZGF '" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Enumerating trisubstituted al kanes " }{XPPEDIT 18 0 "C[n]*H[2*n-1]*X*Y*Z" "*,&%\"CG6#%\"nG\"\"\"&% \"HG6#,&*&\"\"#F'F&F'F'\"\"\"!\"\"F'%\"XGF'%\"YGF'%\"ZGF'" }{TEXT -1 51 " is equivalent to enumerating disubstituted alkyls " }{XPPEDIT 18 0 "C[n]*H[2*n-1]*X*Y" "**&%\"CG6#%\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F'F' \"\"\"!\"\"F'%\"XGF'%\"YGF'" }{TEXT -1 30 ". In this section, we assu me " }{XPPEDIT 18 0 "X" "I\"XG6\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Y " "I\"YG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Z" "I\"ZG6\"" }{TEXT -1 139 " to be distinct. The grammar is more involved than in the dis ubstituted case: we have to distinguish several cases, according to wh ich of " }{XPPEDIT 18 0 "X" "I\"XG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Y" "I\"YG6\"" }{TEXT -1 81 " go into subtrees, and into which su btrees. The corresponding class equation is " }{XPPEDIT 18 0 "S2_Alky l[X,Y]=Carbon*S2_Alkyl[X,Y]*(Epsilon+Alkyl+Alkyl^2)+Carbon*(X+S1_Alkyl [X])*(Y+S1_Alkyl[Y])*(Epsilon+Alkyl)" "/&%)S2_AlkylG6$%\"XG%\"YG,&*(%' CarbonG\"\"\"&F$6$F&F'F+,(%(EpsilonGF+%&AlkylGF+*$F0\"\"#F+F+F+**F*F+, &F&F+&%)S1_AlkylG6#F&F+F+,&F'F+&F66#F'F+F+,&F/F+F0F+F+F+" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "gramm_S2_Alkyl:=S2_A lkyl[X,Y]=Union(Prod(Carbon,S2_Alkyl[X,Y],Set(Alkyl,card<=2)),Prod(Car bon,Union(S1_Alkyl[X],X),Union(S1_Alkyl[Y],Y),Set(Alkyl,card<=1))):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "specs_S2_Alkyl:=[S2_Alkyl [X,Y],\{gramm_S2_Alkyl,gramm_S1_Alkyl,op(subs(X=Y,[gramm_S1_Alkyl])),g ramm_Alkyl\},unlabelled]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "seq(count(specs_S2_Alkyl,size=i),i=0..50);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6U\"\"!\"\"\"\"\"%\"#8\"#U\"$J\"\"$-%\"%=7\"%dO\"&**3\"\" &)HK\"&d_*\"'W)z#\"'!R>)\"(#R#R#\"(cz'p\")u4D?\")*3W(e\"*!)*=,<\"***R \">\\\"+iQ\\?9\"+I0%p4%\",_@<.=\"\",tuDqR$\",6t-yw*\"-J%GVi!G\"-iQs(e0 )\".#*****>4J#\".'>,,nCm\"/esD3*y*=\"/Ha/i0Ma\"0%>P7D.b:\"0$*H;kmwW%\" 1B\\,!)))\\r7\"1f>\"o`:Lj$\"2a'4pg\"zx.\"\"2$R'4-E,I'H\"2'=qi!p^lX)\"3 HPg)36rET#\"3(HOX&eX/\")o\"4\"R?ab;o'='>\"49\"*eR]\\&y\"f&\"5/#Q)yi(pA Lf\"\"5`gq(>D3R(QX\"6*[4QX#)[$fDH\"\"6j@SF0]&)e+o$\"7QaL6c:Ks\\Z5\"76y wJGxQO\"4)H\"7)>J.6q<0H5[)\"83$=`[. " 0 "" {MPLTEXT 1 0 33 "alk:=draw(specs_S2_Alkyl,size=6);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$alkG-%%ProdG6%%'CarbonG-F&6&F(%\"XG-F&6%F(-F&6$-F& 6$F(%\"YG-%$SetG6#-F&6$F(%(EpsilonGF3F8F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "The following procedure rewrites a disubstituted alkyl in to a more readable way." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "nice:=proc(alk) subs(\{[C,X]=CX,[C,Y]=CY,[C,X,Y]=CXY\},eval(subs(\{Ep silon=NULL,Carbon=C,Prod=proc() [args] end,Set=proc() args end\},alk)) ) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nice(alk);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$%\"CG7%F$%\"XG7%F$7$%#CYG7#F$F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "The following procedures compute t he size and height of a given disubstituted alkyl." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 224 "size:=proc(alk) option remember; `if`(nops (alk)=2,1+convert(map(size,op(2,alk)),`+`),1+convert(map(size,[op(2..- 2,alk)]),`+`)+convert(map(size,op(-1,alk)),`+`)) end:\nsize(Carbon):=1 :\nsize(X):=0:\nsize(Y):=0:\nsize(Epsilon):=0:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 223 "height:=proc(alk) option remember; `if`(nops( alk)=2,1+max(op(map(height,op(2,alk)))),1+max(op(map(height,[op(2..-2, alk)])),op(map(height,op(-1,alk))))) end:\nheight(Carbon):=1:\nheight( X):=0:\nheight(Y):=0:\nheight(Epsilon):=0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Here is a disubstituted alkyl with 50 carbon atoms, its n ice representation and height." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "alk:=draw(specs_S2_Alkyl,size=50):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 10 "nice(alk);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7$%\" CG7%F$7$F$7&F$7$F$7$F$7%F$7%F$7$F$7$F$7&F$7&F$7&F$7#%#CXG7$F$7&F$7#F$F 5F57$F$7$F$7$F$F5F87%F$F5F5F5F8F97%F$F8F9F5F57%F$7$%#CYGF6F8" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "height(alk);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Tri substituted alkanes, " }{XPPEDIT 18 0 "C[n]*H[2*n-1]*X[2]*Y" "**&%\"CG 6#%\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F'F'\"\"\"!\"\"F'&%\"XG6#\"\"#F'%\"Y GF'" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Enumerating trisubstituted \+ alkanes " }{XPPEDIT 18 0 "C[n]*H[2*n-1]*X[2]*Y" "**&%\"CG6#%\"nG\"\"\" &%\"HG6#,&*&\"\"#F'F&F'F'\"\"\"!\"\"F'&%\"XG6#\"\"#F'%\"YGF'" }{TEXT -1 51 " is equivalent to enumerating disubstituted alkyls " }{XPPEDIT 18 0 "C[n]*H[2*n-1]*X[2]" "*(&%\"CG6#%\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F 'F'\"\"\"!\"\"F'&%\"XG6#\"\"#F'" }{TEXT -1 30 ". In this section, we \+ assume " }{XPPEDIT 18 0 "X" "I\"XG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Y" "I\"YG6\"" }{TEXT -1 40 " to be distinct. The class equation is " }{XPPEDIT 18 0 "S2_Alkyl[X,X]=Carbon*S2_Alkyl[X,X]*(Epsilon+Alky l+Alkyl^2)+Carbon*(S1_Alkyl[X]^2+S1_Alkyl[X]*X+X^2)*(Epsilon+Alkyl)" " /&%)S2_AlkylG6$%\"XGF&,&*(%'CarbonG\"\"\"&F$6$F&F&F*,(%(EpsilonGF*%&Al kylGF**$F/\"\"#F*F*F**(F)F*,(*$&%)S1_AlkylG6#F&\"\"#F**&&F66#F&F*F&F*F **$F&\"\"#F*F*,&F.F*F/F*F*F*" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "gramm_S2b_Alkyl:=S2_Alkyl[X,X]=Union(Prod(Carbo n,S2_Alkyl[X,X],Set(Alkyl,card<=2)),Prod(Carbon,Union(Prod(S1_Alkyl[X] ,S1_Alkyl[X]),Prod(S1_Alkyl[X],X),Prod(X,X)),Set(Alkyl,card<=1))):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "specs_S2b_Alkyl:=[S2_Alkyl[ X,X],\{gramm_S2b_Alkyl,gramm_S1_Alkyl,gramm_Alkyl\},unlabelled]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "seq(count(specs_S2b_Alkyl,si ze=i),i=0..50);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6U\"\"!\"\"\"\"\"$\" \"*\"#H\"#\"*\"$#G\"$k)\"%CE\"%0z\"&lO#\"&e/(\"'8)3#\"'+kh\"(aL\"=\"(t '=`\")s!fb\"\")R-TX\"*_(eA8\"*4O\\%Q\"+&e2f6\"\"+$>#yLK\"+8RHe$*\",a)* =[q#\",;sV(3y\"-\\pt)>D#\"-VH5E)['\".Q&z'pw'=\".](RPnr`\"/a'Q_pPa\"\"/ (4DyYMV%\"0:!\\fCNs7\"0Y,zI6#\\O\"1(RJ(4Q+Y5\"1)G.!y)yl*H\"1!R#QcS1!e) \"2p4]()G/bX#\"26i<\"*peS-(\"308(>@Nq$3?\"3gr?G)Hy+u&\"4^C;qC83*R;\"4: 4NNM>gLo%\"5b'36&\\Lj-P8\"5eX+\\W$f*p:Q\"6B(Q#)zrAqf)3\"\"6INZ)3v\\1u/ J\"6()GI2WiVsA&))\"77j%*)\\6E'[DBD\"7ZJ9qP\\FQM!>(\"82MK768y*)\\%[?\"8 7!>Y" "0%\"XG%\"YG" }{TEXT -1 2 ")." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Here is an example of a disubstituted alkyl with \+ 6 carbon atoms, obtained by the command " }{HYPERLNK 17 "combstruct[dr aw]" 2 "combstruct[draw]" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "alk:=draw(specs_S2b_Alkyl,size=6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$alkG-%%ProdG6%%'CarbonG-F&6$-F&6%F(-F&6$-F&6$F( %\"XG-%$SetG6#-F&6$F(%(EpsilonG-F36$F5F5F1F7" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 80 "The following procedure rewrites a disubstituted alkyl \+ into a more readable way." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "nice:=proc(alk) subs(\{[C,X]=CX,[C,X,X]=CX[2]\},eval(subs(\{Epsil on=NULL,Carbon=C,Prod=proc() [args] end,Set=proc() args end\},alk))) e nd:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nice(alk);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$%\"CG7$7&F$7$%#CXG7#F$F)F)%\"XG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Here are the 9 disubstituted compo unds " }{XPPEDIT 18 0 "C[n]*H[2*n-1]*X[2]*Y" "**&%\"CG6#%\"nG\"\"\"&% \"HG6#,&*&\"\"#F'F&F'F'\"\"\"!\"\"F'&%\"XG6#\"\"#F'%\"YGF'" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "n=3" "/%\"nG\"\"$" }{TEXT -1 1 ":" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "map(nice,allstructs(specs_S2 b_Alkyl,size=3));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7+7%%\"CG7$7#%#CX G%\"XG7#F%7%F%7$F%7$F)F)F*7$F%7%F%F-F*7$F%7$7$F(F*F)7$F%7$7$F%F'F)7$F% 7$F%F,7$F%7$F%F&7%F%F-7$F%F*7$F%7$F'F'" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 39 "Conclusion: multiply substituted alkyls" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "In the previous sections, we have enumerated th e substituted compounds " }{XPPEDIT 18 0 "C[n]*H[2*n+1]*X" "*(&%\"CG6# %\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F'F'\"\"\"F'F'%\"XGF'" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "C[n]*H[2*n]*X*Y" "**&%\"CG6#%\"nG\"\"\"&%\"HG6#*&\" \"#F'F&F'F'%\"XGF'%\"YGF'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "C[n]*H[2*n -1]*X*Y*Z" "*,&%\"CG6#%\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F'F'\"\"\"!\"\"F '%\"XGF'%\"YGF'%\"ZGF'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "C[n]*H[2*n -1]*X[2]*Y" "**&%\"CG6#%\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F'F'\"\"\"!\"\" F'&%\"XG6#\"\"#F'%\"YGF'" }{TEXT -1 45 ". We could in principle enume rate the class " }{XPPEDIT 18 0 "S[p[1],`...`,p[t]]" "&%\"SG6%&%\"pG6# \"\"\"%$...G&F&6#%\"tG" }{TEXT -1 42 " of compounds obtained after sub stituting " }{XPPEDIT 18 0 "p[1]" "&%\"pG6#\"\"\"" }{TEXT -1 19 " hydr ogen atoms by " }{XPPEDIT 18 0 "X^`(1)`" ")%\"XG%$(1)G" }{TEXT -1 8 " \+ atoms, " }{XPPEDIT 18 0 "p[2]" "&%\"pG6#\"\"#" }{TEXT -1 19 " hydrogen atoms by " }{XPPEDIT 18 0 "X^`(2)`" ")%\"XG%$(2)G" }{TEXT -1 13 " ato ms, ..., " }{XPPEDIT 18 0 "p[t]" "&%\"pG6#%\"tG" }{TEXT -1 19 " hydrog en atoms by " }{XPPEDIT 18 0 "X^`(t)`" ")%\"XG%$(t)G" }{TEXT -1 33 " a toms, and one hydrogen atom by " }{XPPEDIT 18 0 "Y" "I\"YG6\"" }{TEXT -1 73 " (so as to plant the trees). Doing so would require to define \+ the class " }{XPPEDIT 18 0 "S[q[1],`...`,q[t]]" "&%\"SG6%&%\"qG6#\"\" \"%$...G&F&6#%\"tG" }{TEXT -1 10 " for each " }{XPPEDIT 18 0 "q[1]<=p[ 1]" "1&%\"qG6#\"\"\"&%\"pG6#\"\"\"" }{TEXT -1 7 ", ..., " }{XPPEDIT 18 0 "q[t]<=p[t]" "1&%\"qG6#%\"tG&%\"pG6#F&" }{TEXT -1 15 ", and for e ach " }{XPPEDIT 18 0 "q[1]<=p[1]" "1&%\"qG6#\"\"\"&%\"pG6#\"\"\"" } {TEXT -1 7 ", ..., " }{XPPEDIT 18 0 "q[t]<=p[t]" "1&%\"qG6#%\"tG&%\"pG 6#F&" }{TEXT -1 75 ", to write a recursion involving partitions into \+ 4 parts of the multiset \{" }{XPPEDIT 18 0 "X^`(1)`" ")%\"XG%$(1)G" } {TEXT -1 2 " (" }{XPPEDIT 18 0 "q[1]" "&%\"qG6#\"\"\"" }{TEXT -1 14 " \+ times), ..., " }{XPPEDIT 18 0 "X^`(t)`" ")%\"XG%$(t)G" }{TEXT -1 2 " ( " }{XPPEDIT 18 0 "q[t]" "&%\"qG6#%\"tG" }{TEXT -1 20 " times)\}. When the " }{XPPEDIT 18 0 "q[i]" "&%\"qG6#%\"iG" }{TEXT -1 60 "'s are give n, those partitions can be computed by a call to " }{HYPERLNK 17 "comb struct[allstructs]" 2 "combstruct[allstructs]" "" }{TEXT -1 146 ". It follows that we would describe and generate the grammar for multiply \+ substituted alkyls in terms of the grammar for partitions into 4 parts !" }}}}}{MARK "0 2 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 } 6#\"\"\"" }{TEXT -1 7 ", ..., " }{XPPEDIT 18 0 "q[t]<=p[t]" "1&%\"qG6#%\"tG&%\"pG6#F&" }{TEXT -1 15 ", and for e ach " }{XPPEDIT 18 0 "q[1]<=p[1]" "1&%\"qG6#\"\"\"&%\"pG6#\"\"\"" } {TEXT -1 7 ", ..., " }{XPPEDIT 18 0 "q[t]<=p[t]" "1&%\"qG6#%\"tG&%\"pG 6#F&" }{TEXT -1 75 ", to write a recursion involving partitions into \+ 4 parts of the multiset \{" }{XPPEDIT 18 0 "X^`(1)`" ")%\"XG%$(1)G" } {TEXT -1 2 " (" }{XPPEDIT 18 0 "q[1]" "&%\"qG6#\"\"\"" }{TEXT -1 14 " \+ times), ..., " }{XPPEDIT 18 0 "X^`(t)`" ")%\"XG%$(t)G" }{TEXT -1 2 " ( " }{XPPEDIT 18 0 "q[t]" "&%\"qG6#%\"tG" }{TEXT -1 20 " times)\}. When the " }{XPPEDIT 18 0 "q[i]" "&%\"qG6#%\"iG" }{TEXT -1 60 "'s are give n, those partitions can be computed by a call to " }{HYPERLNK 17 "comb struct[allstructs]" 2 "combstruct[allstructs]" "" }{TEXT -1 146 ". It follows that we would describe and generate the grammar for multiply \+ substituted alkyls in terms of the grammar for partitions into 4 parts !" }}}}}{MARK "0 2 crossx Ecrucial }crystalcss.mԡcsstabl mԡctycubic ]cumulxcumulat curr3 a!k׵xcurrent  !d currentbtf x curvyxcustomiz .%fcutcutecuts!kcutsem #'cuttEcx+cxy+cy+cycl(!kxcyclic cyclical cyclotomic!kdagg 2( dataG !>9Dapd!km֌>JECdate.m regexpcounteqnstoasympexpcomputasymptoticexpectatvariancnumbmatchregularexpressgivenbivariatlinearsystemcallsequencsyseqnregparameterlistsetequationnamedescriptfunctreturnelementfirsttermexpansrandomtexttypicalinputwithfunctionworklocaldevelopmneighborhoodherecountsizesolutsolvconstantcoefficientwhildirectcomputatgoesthroughoriginalpolynomialdetailnicodemsalvflajoletmotifstatisticthannualeuropeansymposiumalgorithmesapragujulyexamplwordaargprodatomgautoregexptomatchesgramoverlapaddsymbolicweightgwautogramweighttransformintonumericgeqlcombstructgfeqnunlabellexpectopstddevsqrtalsowith     + !#'2((<)./3m6!>?A9DH+KOQrVW]ab c pde%ffNj!klm7prFsy #qy֌וNԡ}ĦdW׵ J]½NSxE j?SC< )invokOT>jinvokat#3dPQe7pԡNinvolvc*+ !+Kf!kryyĦEipl  iplbtiplcciplfgipltiquox irrational} irrationalit } irregular y irregularit irrelevantisingiso .misolatyEisolatinNisomer+ isomorphic +issacfissu]istingEitalic .mitem QuitemizQiterat ]fiteratedfitself }Eitsxiu!kAgdevgeneralasymptoticexpansglimitcomputatlimitbasedcallsequencexpreqndirparameterexpressequatnameoptionaldirectstraightinversnonnegatintegdescriptfunctcomputwithrespectvariablaboutpointuporderinternalperformassummeanfollowlineexceptrightleftinfinitothercomplexnumberdefaultnumbzerotermdetailthesfunctioncanfoundexamplautomaticsigsambulletinvolaprilmustloadreadliblncotgammasintanalsobner#'body.mԡbodyfootԡbodyheadԡbold.mbond+book+boolean+QapdQi7p֌ԡĦbord .mborel+yborrow}borwein]bose!kboson!kboth_!(.m6]/imFsuy}W׵]xbottommԡ bottommostboundG?+T!k}׵Eboundarf }׵]boxe!kbr!k mxbranchbreach !kbreak 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0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 273 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 274 1 {CSTYLE "" -1 -1 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 48 "Borel Resummation of Dive rgent Series Using Gfun" }}{PARA 19 "" 0 "" {TEXT 256 51 "Fr\351d\351r ic Chyzak, Marianne Durand, and Bruno Salvy\n\n" }{TEXT -1 10 "June, 2 001" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 262 8 "Abstract" }{TEXT -1 7 ": We ex" }{TEXT 265 0 "" }{TEXT -1 343 "pand on ideas of Balser, Lutz, and Sch\344fke by showing how coefficients and integrals involv ed in calculations related to the analytic continuation of Borel trans forms obey simple recurrences that lead to efficient numerical computa tions. This work is a follow-up to a talk by Donald Lutz at our Algor ithms seminar, and summarized in [Durand]." }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 33 "1. Borel-Laplace resummation and " }{TEXT 261 0 "" } {TEXT -1 9 "Euler acc" }{TEXT 260 0 "" }{TEXT -1 9 "eleration" }} {PARA 0 "" 0 "" {TEXT -1 108 "Starting with a linear differential equa tion with polynomial coefficients satisfied by a formal power series" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "x^~ = Sum(x[n]*z^n,n = 0 .. infinit y);" "6#/)%\"xG%\"|irG-%$SumG6$*&&F%6#%\"nG\"\"\")%\"zGF-F./F-;\"\"!%) infinityG" }{TEXT 257 0 "" }{TEXT 258 0 "" }{TEXT 259 0 "" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 87 "it is possible to compute a diffe rential equation satisfied by the Borel transform of " }{XPPEDIT 18 0 "x^~;" "6#)%\"xG%\"|irG" }{TEXT -1 17 ". We assume that " }{XPPEDIT 18 0 "x^~;" "6#)%\"xG%\"|irG" }{TEXT -1 31 " is Gevrey 1, which means \+ that " }{XPPEDIT 18 0 "x[n] <= A*c^n*n!;" "6#1&%\"xG6#%\"nG*(%\"AG\"\" \")%\"cGF'F*-%*factorialG6#F'F*" }{TEXT -1 21 " for some constants " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "c;" "6#%\"cG" }{TEXT -1 34 ", so that the Borel transform of " }{XPPEDIT 18 0 "x^~;" "6#)%\"xG%\"|irG" }{TEXT -1 11 " defined by" }}{PARA 257 " " 0 "" {XPPEDIT 18 0 "y(z) = Sum(x[n]*z^n/n!,n = 0 .. infinity);" "6#/ -%\"yG6#%\"zG-%$SumG6$*(&%\"xG6#%\"nG\"\"\")F'F/F0-%*factorialG6#F/!\" \"/F/;\"\"!%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 121 "is convergent on some neighbourhood of the origin. The Borel tran sform has an \"inverse\", the Laplace transform defined by" }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Laplace(y) = Int(exp(-t/z)* y(t),t = 0 .. infinity);" "6#/-%(LaplaceG6#%\"yG-%$IntG6$*&-%$expG6#,$ *&%\"tG\"\"\"%\"zG!\"\"F4F2-F'6#F1F2/F1;\"\"!%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 62 "Provided this integral converges, t he function it defines has " }{XPPEDIT 18 0 "z*x^~;" "6#*&%\"zG\"\"\") %\"xG%\"|irGF%" }{TEXT -1 18 " for expansion as " }{XPPEDIT 18 0 "proc (z) options operator, arrow; 0 end;" "6#R6#%\"zG7\"6$%)operatorG%&arr owG6\"\"\"!F*F*F*" }{TEXT -1 42 " +. Then applying the change of vari able " }{XPPEDIT 18 0 "t = phi(alpha);" "6#/%\"tG-%$phiG6#%&alphaG" } {TEXT -1 69 " to the integral computes the acceleration \"a la Euler\" [Lutz et al.]" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z*x ^~ = Int(exp(-t/z)*y(phi(psi(t))),t = 0 .. infinity);" "6#/*&%\"zG\"\" \")%\"xG%\"|irGF&-%$IntG6$*&-%$expG6#,$*&%\"tGF&F%!\"\"F4F&-%\"yG6#-%$ phiG6#-%$psiG6#F3F&/F3;\"\"!%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 7 "We note" }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "y(phi(z)) = Sum(d[n]*z^n,n = 0 .. infinity);" "6#/-%\"yG6#-%$phiG6#%\"zG-%$SumG 6$*&&%\"dG6#%\"nG\"\"\")F*F2F3/F2;\"\"!%)infinityG" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "q[n](z) = Int(exp(-t/z)*psi(t)^n,t = 0 .. infinity); " "6#/-&%\"qG6#%\"nG6#%\"zG-%$IntG6$*&-%$expG6#,$*&%\"tG\"\"\"F*!\"\"F 6F5)-%$psiG6#F4F(F5/F4;\"\"!%)infinityG" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 53 " is the functional inverse of the rational function " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 47 ". In terms of formal power series , the product " }{XPPEDIT 18 0 "z*x^~;" "6#*&%\"zG\"\"\")%\"xG%\"|irGF %" }{TEXT -1 31 " equals the Taylor expansion of" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "Sum(q[n]*d[n],n = 0 .. infinity);" "6#-%$SumG6$*&&%\"qG 6#%\"nG\"\"\"&%\"dG6#F*F+/F*;\"\"!%)infinityG" }{TEXT -1 2 " ," }} {PARA 0 "" 0 "" {TEXT -1 20 "where the integrals " }{XPPEDIT 18 0 "q[n ];" "6#&%\"qG6#%\"nG" }{TEXT -1 21 " are independent of " }{XPPEDIT 18 0 "x^~;" "6#)%\"xG%\"|irG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "This process is illustrated in \+ the present worksheet using a simple mapping function " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 153 " on the double confluent Heun equat ion. The Heun equation is the generic differential equation with four \+ regular singular points located at 0, 1, c, and " }{XPPEDIT 18 0 "infi nity;" "6#%)infinityG" }{TEXT -1 134 ", see [DuLoRi92]. The double con fluent Heun equation is obtained by letting the singularity located a t c tend to the one located at " }{XPPEDIT 18 0 "infinity;" "6#%)infi nityG" }{TEXT -1 127 ", and the singularity located at 1 tend to 0. T he equation obtained then has two irregular \nsingular points located at 0 and " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 77 ". The example we study is the double confluent Heun equation in the f orm " }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "heun_infp := z^2*di ff(f(z),z,z)+(z+alpha*z^2+alpha)*diff(f(z),z)+(2*alpha*z^2*beta[1]+alp ha*z^2+alpha^2*z-2*gamma*z+2*alpha*beta[-1]-alpha)*f(z)/(2*z):" "6#>%* heun_infpG,(*&%\"zG\"\"#-%%diffG6%-%\"fG6#F'F'F'\"\"\"F/*&,(F'F/*&%&al phaGF/*$F'\"\"#F/F/F3F/F/-F*6$-F-6#F'F'F/F/*(,.**\"\"#F/F3F/F'\"\"#&%% betaG6#\"\"\"F/F/*&F3F/*$F'\"\"#F/F/*&F3\"\"#F'F/F/*(\"\"#F/%&gammaGF/ F'F/!\"\"*(\"\"#F/F3F/&F@6#,$\"\"\"FKF/F/F3FKF/-F-6#F'F/*&\"\"#F/F'F/F KF/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "readlib(gfun):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Since the point of interest is inf inity and the " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 66 " packag e works at the origin, we first change the variable (using " } {HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 2 "):" }{TEXT 267 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "heunp:=gfun[algebraicsubs](h eun_infp,z*f-1,f(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&heunpG,(*& ,.*()%\"zG\"\"#\"\"\"%&alphaG\"\"\"&%%betaG6#!\"\"F.!\"#*&F-F,F)F,F.*& )F-F+F,F*F.F2*&%&gammaGF.F*F,F+*&F-F,&F06#F.F.F3F-F2F.-%\"fG6#F*F.F.*& ,(*$F)F,F3*&)F*\"\"$F,F-F,F+*&F*F,F-F,F+F.-%%diffG6$F " 0 "" {XPPEDIT 19 1 "paramform:=[alpha=-1,beta[-1]= 1/2,beta[1]=1/2,gamma=1/3]:" "6#>%*paramformG7&/%&alphaG,$\"\"\"!\"\"/ &%%betaG6#,$\"\"\"F**&\"\"\"\"\"\"\"\"#F*/&F-6#\"\"\"*&\"\"\"F3\"\"#F* /%&gammaG*&\"\"\"F3\"\"$F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Sub stitution of these parameters in the differential equation gives" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "heun:=subs(paramform,heunp); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%heunG,(*&,&%\"zG#!\"\"\"\"$\"\" #\"\"\"F--%\"fG6#F(F-F-*&,(*$)F(F,\"\"\"!\"#*$)F(F+F5F6F(F6F--%%diffG6 $F.F(F-F-*&F8F5-F:6$F.-%\"$G6$F(F,F-F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "From this equation, we obtain a recurrence equation for t he Taylor series coefficients" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "recheunseries:=gfun[diffeqtorec](heun,f(z),u(n));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%.recheunseriesG<$,(*&%\"nG\"\"\"-%\"uG6#F(F)! \"'*&,(!\"(F)F(!#7*$)F(\"\"#\"\"\"F-F)-F+6#,&F(F)F)F)F)F)*&,&F-F)F(F-F )-F+6#,&F(F)F4F)F)F)/-F+6#\"\"!FA" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "This recurrence yields an efficient procedure to evaluate the coef fici" }{TEXT 266 0 "" }{TEXT -1 17 "ents recursively:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "heundiv:=gfun[rectoproc](recheunser ies union \{u(1)=1/2\},u(n),remember);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(heundivGR6#%\"nG6\"6#%)rememberGE\\s#\"\"\"#F,\"\"#\"\"!F/,$* &,(-9!6#,&9$F,!\"#F,\"#7-F46#,&F7F,!\"\"F,!\"(*&,(F3!\"'F:F9*&F:F,F7F, FAF,F7\"\"\"F,FC,&\"\"'F,F7FA!\"\"F=F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "From this procedure, the divergence is clear from the gro wth of the first coefficients:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "seq(heundiv(i),i=1..15);" }}{PARA 12 "" 1 "" {XPPMATH 20 "61# \"\"\"\"\"##!\"(\"#7#\"$R\"\"$W\"#!%Pm\"%#f##\"'`()f\"&3A'#!)\"z)e')\" (Si'=#\",Z**)3D=\")SY=n#!.H7Y$zr_\"+!)[v@G#\"0^45;%\\`G\",?j<\\$>#!3V. Z`t%*41'*\".g*['))RJ(#\"6JV1$)z2E?]r&\"0+w$*=$R)Q%#!9PQ`%>@#[rxB?T\"2+ ;))\\]Rj*G#\" " 0 "" {XPPEDIT 19 1 "calculheundiv:=proc(heundiv,z)\nlocal total,previous ,last,n;\nprevious:=heundiv(1)*z;total:=previous;last:=heundiv(2)*z^2; \nfor n from 3 while abs(previous)>abs(last) do\ntotal:=total+last;\np revious:=last;\nlast:= heundiv(n)*z^n od;\nuserinfo(1,'heundiv',n,last );\nevalf(total)\nend:" "6#>%.calculheundivGR6$%(heundivG%\"zG7&%&tota lG%)previousG%%lastG%\"nG6\"F.C(>F+*&-F'6#\"\"\"\"\"\"F(F5>F*F+>F,*&-F '6#\"\"#F5*$F(\"\"#F5?(F-\"\"$F5F.2-%$absG6#F,-FB6#F+C%>F*,&F*F5F,F5>F +F,>F,*&-F'6#F-F5)F(F-F5-%)userinfoG6&\"\"\".F'F-F,-%&evalfG6#F*F.F.F. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot('calculheundiv'(he undiv,z),z=0..0.3);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7]p7$ \"\"!F(7$$\"+]i9Rl!#7$\"+w[*[C$F,7$$\"+WA)GA\"!#6$\"+Pw))GgF,7$$\"+Qeu i=F2$\"+FMF<\"*F,7$$\"+j3&o]#F2$\"+(3&>=7F27$$\"+pX*y9$F2$\"+#*)G*=:F2 7$$\"+WTAUPF2$\"+[t.%z\"F27$$\"+%*zhdVF2$\"+f0Bv?F27$$\"+%>fS*\\F2$\"+ 1d@iBF27$$\"+>$f%GcF2$\"+\"[0Xk#F27$$\"+Dy,\"G'F2$\"+:&R5$HF27$$\"+8LZF27$$\"+/QBE6Fcp$\"+X=k,]F27$$\"+:o?&=\" Fcp$\"+8G!\\B&F27$$\"+a&4*\\7Fcp$\"+sWo)[&F27$$\"+j=_68Fcp$\"+jwt6dF27 $$\"+Wy!eP\"Fcp$\"+[h&=&fF27$$\"+UC%[V\"Fcp$\"+BCPohF27$$\"+J#>&)\\\"F cp$\"+6InWkF27$$\"+>:mk:Fcp$\"+Fn/'p'F27$$\"+w&QAi\"Fcp$\"+t-r9pF27$$ \"+v4L`;Fcp$\"+JT!H.(F27$$\"+uLU%o\"Fcp$\"+&f#G^rF27$$\"+ZPX#p\"Fcp$\" +1w*==(F27$$\"+?T[+?sF27$$\"+2$* \\/luF27$$\"+M aKs=Fcp$\"+Z0Fcp$\"+h[LjyF27$$\"+:K^+?Fcp$\"+(eMI\"y?Fcp$\"+QgCL')F27$$\"+)[k*z?Fcp$\"+* GD-k)F27$$\"+v;I(3#Fcp$\"+as:o')F27$$\"+i)QY4#Fcp$\"+bO6'p)F27$$\"+OKJ 4@Fcp$\"+*o-@v)F27$$\"+4w)R7#Fcp$\"+0#)>3))F27$$\"+WN2c@Fcp$\"+lBDJ*)F 27$$\"+y%f\")=#Fcp$\"+rp!\\0*F27$$\"+/-a[AFcp$\"+:OY*G*F27$$\"+ial6BFc p$\"+vdlP&*F27$$\"+j@OtBFcp$\"+(*[!Qy*F27$$\"+fL'zV#Fcp$\"+t)zX+\"Fcp7 $$\"+!*>=+DFcp$\"+b#*GI5Fcp7$$\"+E&4Qc#Fcp$\"+g(Hr0\"Fcp7$$\"+g)f`f#Fc p$\"+&3m12\"Fcp7$$\"+%>5pi#Fcp$\"+!*[O%3\"Fcp7$$\"+NfSTEFcp$\"+fdr!4\" Fcp7$$\"+v;!fl#Fcp$\"+cP5(4\"Fcp7$$\"+WOrdEFcp$\"+7\\!z4\"Fcp7$$\"+6c_ fEFcp$\"+lmq)4\"Fcp7$$\"+zvLhEFcp$\"+nb45(*F27$$\"+Y&\\Jm#Fcp$\"+jYu8( *F27$$\"+\"[tnm#Fcp$\"+5^-@(*F27$$\"+;uRqEFcp$\"+4=GG(*F27$$\"+'GXwn#F cp$\"++QsU(*F27$$\"+bJ*[o#Fcp$\"+a-2d(*F27$$\"+k17=FFcp$\"+Jig@)*F27$$ \"+r\"[8v#Fcp$\"+Yd3%))*F27$$\"+Ijy5GFcp$\"+ZKf!***F27$$\"+/)fT(GFcp$ \"+(Q\\'45Fcp7$$\"+1j\"[$HFcp$\"+j#=!>5Fcp7$$\"\"$!\"\"$\"+](=#G5Fcp-% 'COLOURG6&%$RGBG$\"#5FddlF(F(-%+AXESLABELSG6$Q\"z6\"%!G-%%VIEWG6$;F(Fb dl%(DEFAULTG" 2 376 376 376 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 -22808 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "As " }{XPPEDIT 18 0 "z;" " 6#%\"zG" }{TEXT -1 68 " tends to infinity, the imprecision of this sum mation grows quickly." }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 57 "2. Diffe rential equation satisfied by the Borel transform" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 116 "The class of solutions of linear differential equ ations enjoys many closure properties which are implemented in the " } {HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 225 " package for the case o f equations with polynomial coefficients. One of them is closure under Borel transform. Here is the differential equation satisfied by the B orel transform of the divergent solution of the Heun equation:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "bdeqp:=op(select(has,gfun[bo rel](heunp,f(z),'diffeq'),z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&b deqpG,**&,&*&%&alphaG\"\"\"&%%betaG6#!\"\"F*\"\"#F)F.F*-%\"fG6#%\"zGF* F**&,**&F3F*F)\"\"\"!\"#*$)F)F/F7F*F/F*%&gammaGF8F*-%%diffG6$F0F3F*F** &,(F3\"\"'*&F)F7&F,6#F*F*F/F)!\"$F*-F=6$F0-%\"$G6$F3F/F*F**&,&*$)F3F/F 7F/F6F8F*-F=6$F0-FI6$F3\"\"$F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "and the equation specialized at the parameters" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "bdeq:=subs(paramform,bdeqp):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "From this follows the recurrence satisfie d by the Taylor coefficients of the Borel transform:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "collect(gfun[diffeqtorec](bdeqp,f(z),a(n) ),a,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(%&alphaG\"\"\",(&% %betaG6#!\"\"!\"#F&F&%\"nG\"\"#F&-%\"aG6#F-F&F+*(,&F-F&F&F&F&,,*$)F-F. \"\"\"F.F-\"\"%*$)F%F.F7F&F.F&%&gammaGF,F&-F06#F3F&F&*,F%F7F3F7,&F-F&F .F&F&,(F-F.&F)6#F&F,\"\"$F&F&-F06#F?F&F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "l15:= gfun[rectoproc](\{subs(paramform,%),a(0)=0,a(1)=1/2\},a(n),list)(15); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$l15G72\"\"!#\"\"\"\"\"##!\"(\"# C#\"$R\"\"$k)#!%Pm\"&3A'#\"'`()f\"(g\\Y(#!)\"z)e')\"++GpV8#\",Z**)3D= \"-+ce5'Q$#!.H7Y$zr_\"0+;wcJx8\"#\"0^45;%\\`G\"1+;+.\"H9-(#!3V.Z`t%*41 '*\"5+![gM4?+Tl##\"6JV1$)z2E?]r&\"8++o\"RohKhq^<#!9PQ`%>@#[rxB?T\";++c 5iOD.d7N(Q\"#\"++;%3tFaCew2c)H\"#!?Fb!Htc\\\\$*HrRQ3 d$\"A++s-z?32S!G!o$G!=9#\"B$3lJrLh,5u(H=%[M\"=%\"D++?Fu&>M#*[I`rVdr'y \"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "This list corresponds to th e list above, the " }{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 27 "th ele ment being scaled by " }{XPPEDIT 18 0 "1/k!;" "6#*&\"\"\"\"\"\"-%*fact orialG6#%\"kG!\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 147 "This differential equation and this recurrence can be used to \+ compute (but not necessarily fast) the analytic continuation of the Bo rel transform. " }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 93 "3. The coeffic ients of the composition with an algebraic function satisfy a linear r ecurrence" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "This is another clo sure property of solutions of linear differential equation that we exp loit here. The coefficients " }{XPPEDIT 18 0 "d[n];" "6#&%\"dG6#%\"nG " }{TEXT -1 15 " are defined by" }}{PARA 262 "" 0 "" {XPPEDIT 18 0 "y( phi(z)) = Sum(d[n]*z^n,n = 0 .. infinity);" "6#/-%\"yG6#-%$phiG6#%\"zG -%$SumG6$*&&%\"dG6#%\"nG\"\"\")F*F2F3/F2;\"\"!%)infinityG" }}{PARA 0 " " 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 28 " is a rational function and " }{XPPEDIT 18 0 "y;" "6#%\"yG" } {TEXT -1 15 " is defined by" }}{PARA 263 "" 0 "" {XPPEDIT 18 0 "y(z) \+ = Sum(x[n]*z^n/n!,n = 0 .. infinity);" "6#/-%\"yG6#%\"zG-%$SumG6$*(&% \"xG6#%\"nG\"\"\")F'F/F0-%*factorialG6#F/!\"\"/F/;\"\"!%)infinityG" } {TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 42 "On the example of the H eun equation, with " }{XPPEDIT 18 0 "phi = 1/((1-t)^2)-1;" "6#/%$phiG, &*&\"\"\"\"\"\"*$,&\"\"\"F(%\"tG!\"\"\"\"#F-F(\"\"\"F-" }{TEXT -1 10 " we obtain" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "eq:=(1-z)^2*(f+ 1)-1:" "6#>%#eqG,&*&,&\"\"\"\"\"\"%\"zG!\"\"\"\"#,&%\"fGF)\"\"\"F)F)F) \"\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "dneqp:=gfun[alg ebraicsubs](bdeqp,eq,f(z));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&dneq pG,**&,&%&alphaG\"\"%*&F(\"\"\"&%%betaG6#!\"\"F+!\")F+-%\"fG6#%\"zGF+F +*&,hn!\"%F+%&gammaGF)*&F(\"\"\"&F-6#F+F+\"\"'F4\"#[*&)F(\"\"#F:)F4FAF :!\"'*&F(F:FBF:!$n#F(!\"**&F@F:F4F+F=*&F8F+F4F:!#7*$FBF:!$)>*(F(F:FBF: F;F:\"#g*&F8F:FBF:\"#7*&)F4\"\"$F:F(F:\"$p%*&F4F:F(F:\"#$)*$FQF:\"$1%* (F(F:F;F:F4F:!#I*&)F4F)F:F(F:!$&\\*&)F4\"\"&F:F(F:\"$4$*&)F4F=F:F(F:!$ 0\"*&F@F:FQF:FA*&)F4\"\"(F:F(F:\"#:*$FenF:!$l%*$FhnF:\"$.$*$F\\oF:F]o* $F`oF:Fbo*(F(F:FQF:F;F:!#g*&F8F:FQF:F7*(F(F:FhnF:F;F:FC*(FenF:F(F:F;F: \"#I*$F@F:!\"#F+-%%diffG6$F1F4F+F+*&,RF9F`pF4FIFD\"$i\"F(FRFJ\"#!*FLFY FP!$%QFT!#OFV!$w#FXFOFZ\"$S&Fgn!$o%F[o\"$Y#*&)F4\"\")F:F(F:\"\"**$F_qF :FaqF_o!#sFco\"$`%Feo!$K%Fgo\"$S#FhoFcqFio\"#SF\\pFOF]pFY*(F(F:F\\oF:F ;F:F`pF+-Fbp6$F1-%\"$G6$F4FAF+F+*&,DFD!#:FJF7FP\"#\\FTFAFV\"#CFZ!#\"*F gn\"$0\"F[o!#x*$)F4FaqF:F+F^qFFFbqFFF_o\"#NFco!#hFeo\"#&)Fgo!#qFho\"#M *&FgrF:F(F:F+F+-Fbp6$F1-F\\r6$F4FRF+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "and the equation when the parameters are specialized:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "dneq:=gfun[algebraicsubs](b deq,eq,f(z));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%dneqG,(*&,,*$)%\"z G\"\"%\"\"\"\"\"**$)F*\"\"$F,!#O*$)F*\"\"#F,\"#hF*!#]F+\"\"\"F7-%%diff G6$-%\"fG6#F*F*F7F7*&,.F.\"$T\"F(!#v*$)F*\"\"&F,\"#:F2!$B\"F*\"#[!\"'F 7F7-F96$F;-%\"$G6$F*F4F7F7*&,.F(\"#UF.!#[FB!#=*$)F*\"\"'F,F0F2\"#FF*FH F7-F96$F;-FL6$F*F0F7F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "The lin ear recurrence satisfied by the " }{XPPEDIT 18 0 "d[n];" "6#&%\"dG6#% \"nG" }{TEXT -1 18 " follows directly" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "dnRec:=op(select(has,gfun[diffeqtorec](dneq,f(z),a(n) ),n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&dnRecG,.*&,&*$)%\"nG\"\"# \"\"\"\"\"'*$)F*\"\"$F,F0\"\"\"-%\"aG6#F*F1F1*&,*F*!#$*!#OF1F(!#vF.!#= F1-F36#,&F*F1F1F1F1F1*&,*F*\"$o&\"$/%F1F(\"$n#F.\"#UF1-F36#,&F*F1F+F1F 1F1*&,*F*!%$>\"!%w6F1F(!$6%F.!#[F1-F36#,&F*F1F0F1F1F1*&,*F*\"%U5\"%S7F 1F(\"$\"HF.\"#FF1-F36#,&F*F1\"\"%F1F1F1*&,*F(!#yF*!$O$!$![F1F.!\"'F1-F 36#,&F*F1\"\"&F1F1F1" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 17 "4. The in tegrals " }{XPPEDIT 18 0 "q[n];" "6#&%\"qG6#%\"nG" }{TEXT -1 28 " sati sfy a linear recurrence" }}{PARA 0 "" 0 "" {TEXT -1 68 "The property a bove does not depend on the specific divergent series " }{XPPEDIT 18 0 "x^~;" "6#)%\"xG%\"|irG" }{TEXT -1 80 " that one is resumming. This \+ allows one to precompute efficiently the integrals " }{XPPEDIT 18 0 "q [n](z);" "6#-&%\"qG6#%\"nG6#%\"zG" }{TEXT -1 28 " given the mapping fu nction " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 173 ". Indeed the \+ general theory of holonomic function has recently led to symbolic summ ation and integration algorithms that turn out to apply to the integra l representation of " }{XPPEDIT 18 0 "q[n](z);" "6#-&%\"qG6#%\"nG6#%\" zG" }{TEXT -1 224 ". The goal of these algorithms is to derive (system s of) linear functional equations, differential or difference, satisfi ed by a sum or an integral. We now proceed to use a prototypical imple mentation of them in the package " }{HYPERLNK 17 "Mgfun" 2 "Mgfun" "" }{TEXT -1 31 " to obtain a recurrence on the " }{XPPEDIT 18 0 "q[n](z) ;" "6#-&%\"qG6#%\"nG6#%\"zG" }{TEXT -1 138 ". Then, we prove a theorem that by-passes the general theory of holonomic functions, and recover the same recurrence in a more direct way." }}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 32 "(This section uses a version of " }{HYPERLNK 17 "Mgfun " 2 "Mgfun" "" }{TEXT -1 30 " that is not distributed yet.)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "readlib(Mgfun):" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "phi:=1/(1-t)^2-1:" "6#>%$phiG,&*&\"\"\"\"\"\" *$,&\"\"\"F(%\"tG!\"\"\"\"#F-F(\"\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "F:=exp(-phi/z)*t^n*diff(phi,t):" "6#>%\"FG*(-%$expG6#,$ *&%$phiG\"\"\"%\"zG!\"\"F.F,)%\"tG%\"nGF,-%%diffG6$F+F0F," }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "ct:=Mgfun[creative_telescoping](F,n:: shift,t::diff);" "6#>%#ctG-&%&MgfunG6#%5creative_telescopingG6%%\"FG'% \"nG%&shiftG'%\"tG%%diffG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ctG7$, **&,&%\"zG\"\"\"*&%\"nGF*F)F*F*F*-%#_fG6$F,%\"tGF*F**&,&F)\"\"$F+F3F*- F.6$,&F,F*\"\"#F*F0F*F**&,&F+!\"\"F)F:F*-F.6$,&F,F*F3F*F0F*F**&,(F+!\" $F)F@!\"#F*F*-F.6$,&F,F*F*F*F0F*F***F)\"\"\"F0F*,*F:F*F0F3*$)F0F7FFF@* $)F0F3FFF*F*F-FF" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "h[n](t) :=%[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%\"hG6#%\"nG6#%\"tG,**&, &%\"zG\"\"\"*&F(F/F.F/F/F/-%#_fG6$F(F*F/F/*&,&F.\"\"$F0F6F/-F26$,&F(F/ \"\"#F/F*F/F/*&,&F0!\"\"F.F=F/-F26$,&F(F/F6F/F*F/F/*&,(F0!\"$F.FC!\"#F /F/-F26$,&F(F/F/F/F*F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "H[n](t):=%%[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%\"HG6#%\"nG6#% \"tG**%\"zG\"\"\"F*F-,*!\"\"F-F*\"\"$*$)F*\"\"#\"\"\"!\"$*$)F*F0F4F-F- -%#_fG6$F(F*F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "The meaning of \+ the previous computation is that the differential equation" }}{PARA 272 "" 0 "" {XPPEDIT 18 0 "h[n](t)+diff(H[n](t),t) = 0;" "6#/,&-&%\"hG 6#%\"nG6#%\"tG\"\"\"-%%diffG6$-&%\"HG6#F)6#F+F+F,\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 91 "holds. This can be viewed as a differential-differen ce relation satisfied by the integrand " }{XPPEDIT 18 0 "f[n](z,t);" " 6#-&%\"fG6#%\"nG6$%\"zG%\"tG" }{TEXT -1 27 ". Now, integrating between " }{XPPEDIT 18 0 "-1;" "6#,$\"\"\"!\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "1;" "6#\"\"\"" }{TEXT -1 64 " returns a non-homogeneous differential equation in the integral" }}{PARA 273 "" 0 "" {XPPEDIT 18 0 "q[n](z) = int(f[n](z,t),t = -1 .. 1);" "6#/-&%\"qG6#%\"nG6#%\"zG -%$intG6$-&%\"fG6#F(6$F*%\"tG/F3;,$\"\"\"!\"\"\"\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "namely" }}{PARA 271 "" 0 "" {XPPEDIT 18 0 "int(h[n](t),t = -1 .. 1)+H[n](1)-H[n](-1) = 0;" "6#/,(-%$intG6$-&% \"hG6#%\"nG6#%\"tG/F.;,$\"\"\"!\"\"\"\"\"\"\"\"-&%\"HG6#F,6#\"\"\"F5-& F86#F,6#,$\"\"\"F3F3\"\"!" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 40 "where the integral rewrites in terms of " }{XPPEDIT 18 0 "q[n](z); " "6#-&%\"qG6#%\"nG6#%\"zG" }{TEXT -1 3 " as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Int(h[n](t),t=-1..1)=eval(subs(_f=unapply(q[n](z ),n,t),ct[1]));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,**&,&%\" zG\"\"\"*&%\"nGF+F*F+F+F+-%#_fG6$F-%\"tGF+F+*&,&F*\"\"$F,F4F+-F/6$,&F- F+\"\"#F+F1F+F+*&,&F,!\"\"F*F;F+-F/6$,&F-F+F4F+F1F+F+*&,(F,!\"$F*FA!\" #F+F+-F/6$,&F-F+F+F+F1F+F+/F1;F;F+,**&F)\"\"\"-&%\"qG6#F-6#F*F+F+*&F3F J-&FM6#F7FOF+F+*&F:FJ-&FM6#F>FOF+F+*&F@FJ-&FM6#FEFOF+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "As to the non-homogeneous part " } {XPPEDIT 18 0 "H[n](1)-H[n](-1);" "6#,&-&%\"HG6#%\"nG6#\"\"\"\"\"\"-&F &6#F(6#,$\"\"\"!\"\"F2" }{TEXT -1 59 ", we readily evaluate it, verify ing that it is 0 by chance." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "H[n](t):=factor(eval(subs(_f=unapply(F,n,t),H[n](t))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%\"HG6#%\"nG6#%\"tG**%\"zG\"\"\"F*F-),&! \"\"F-F*F-\"\"$\"\"\"%\"fGF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "assume(z>0); assume(n,integer); factor(simplify(limit(op(2,%),t =1)-limit(op(2,%),t=-1))): non_hom:=subs([z='z',n='n'],%); z:='z': n:= 'n':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(non_homG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Consequently, we have obtained the follow ing recurrence on the integrals " }{XPPEDIT 18 0 "q[n](z);" "6#-&%\"qG 6#%\"nG6#%\"zG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "collect(eval(subs(_f=unapply(q[n](z),n,t),ct[1])),q,factor)=0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&,(*&%\"nG\"\"\"%\"zGF)!\"$F*F +!\"#F)F)-&%\"qG6#,&F(F)F)F)6#F*F)F)*(F*\"\"\"F1F)-&F/6#,&F(F)\"\"$F)F 2F)!\"\"*(F*F4F1F4-&F/6#,&F(F)\"\"#F)F2F)F9*(F*F4F1F4-&F/6#F(F2F)F)\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "collect(subs(n=n-4,% ),q,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**(%\"zG\"\"\",&%\"n GF'!\"$F'F'-&%\"qG6#,&F)F'!\"\"F'6#F&F'F0*(F&\"\"\"F(F3-&F-6#,&F)F'!\" #F'F1F'\"\"$*(F&F3F(F3-&F-6#,&F)F'!\"%F'F1F'F'*&,(*&F)F'F&F3F*F&\"\"*F 8F'F'-&F-6#F(F1F'F'\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "More \+ generally, a differential equation with respect to " }{XPPEDIT 18 0 "z ;" "6#%\"zG" }{TEXT -1 103 ", or even a system of mixed differential-d ifference equations could be obtained by the same algorithms." }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "The following result had not been noticed by Lutz et al., but might prove useful in numerical computati ons." }}{PARA 0 "" 0 "" {TEXT 263 7 "Theorem" }{TEXT -1 32 ". With the above notations, let " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 266 " " 0 "" {XPPEDIT 18 0 "sum(p[k](n)*a(n+k),k = 0 .. K) = 0;" "6#/-%$sumG 6$*&-&%\"pG6#%\"kG6#%\"nG\"\"\"-%\"aG6#,&F.F/F,F/F//F,;\"\"!%\"KGF6" } }{PARA 0 "" 0 "" {TEXT -1 150 "be the linear recurrence satisfied by t he Taylor coefficients at the origin of a power series solution of the first-order linear differential equation" }}{PARA 267 "" 0 "" {XPPEDIT 18 0 "diff(G(t),t) = (diff(phi(t),t,t)/diff(phi(t),t)-diff(ph i(t),t)/z)*G(t);" "6#/-%%diffG6$-%\"GG6#%\"tGF**&,&*&-F%6%-%$phiG6#F*F *F*\"\"\"-F%6$-F16#F*F*!\"\"F3*&-F%6$-F16#F*F*F3%\"zGF8F8F3-F(6#F*F3" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "Then the integrals " } {XPPEDIT 18 0 "q[n];" "6#&%\"qG6#%\"nG" }{TEXT -1 23 " satisfy the rec urrence" }}{PARA 268 "" 0 "" {XPPEDIT 18 0 "sum(p[k](-n)*q[n-k-1](z),k = 0 .. K) = 0;" "6#/-%$sumG6$*&-&%\"pG6#%\"kG6#,$%\"nG!\"\"\"\"\"-&% \"qG6#,(F/F1F,F0\"\"\"F06#%\"zGF1/F,;\"\"!%\"KGF<" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT 264 6 "Proof." }{TEXT -1 78 " The differential e quation in the statement above is satisfied by the function" }}{PARA 269 "" 0 "" {XPPEDIT 18 0 "e^(-phi(u)/z)*diff(phi(u),u);" "6#*&)%\"eG, $*&-%$phiG6#%\"uG\"\"\"%\"zG!\"\"F.F,-%%diffG6$-F)6#F+F+F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "Since the integrals " }{XPPEDIT 18 0 "q[n](z);" "6#-&%\"qG6#%\"nG6#%\"zG" }{TEXT -1 11 " rewrite as" } }{PARA 270 "" 0 "" {XPPEDIT 18 0 "int(e^(-phi(u)/z)*u^n*diff(phi(u),u) ,u = 0 .. 1);" "6#-%$intG6$*()%\"eG,$*&-%$phiG6#%\"uG\"\"\"%\"zG!\"\"F 1F/)F.%\"nGF/-%%diffG6$-F,6#F.F.F//F.;\"\"!\"\"\"" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 114 "by integration by parts and differentiat ion under the integral sign, they satisfy the announced linear recurre nce." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "The following one-line pr ocedure computes a recurrence on the integral" }}{PARA 264 "" 0 "" {XPPEDIT 18 0 "q[n](z) = Int(exp(-t/z)*psi(t)^n,t = 0 .. infinity);" " 6#/-&%\"qG6#%\"nG6#%\"zG-%$IntG6$*&-%$expG6#,$*&%\"tG\"\"\"F*!\"\"F6F5 )-%$psiG6#F4F(F5/F4;\"\"!%)infinityG" }}{PARA 0 "" 0 "" {TEXT -1 6 "wh ere " }{XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 50 " is the functiona l inverse of a rational function " }{XPPEDIT 18 0 "phi;" "6#%$phiG" } {TEXT -1 20 ". It takes as input " }{XPPEDIT 18 0 "phi(t),t,g,n,z;" "6 '-%$phiG6#%\"tGF&%\"gG%\"nG%\"zG" }{TEXT -1 115 " where all the argume nts except the first one are symbols that appear in the output linear \+ recurrence relating the " }{XPPEDIT 18 0 "q[n](z);" "6#-&%\"qG6#%\"nG6 #%\"zG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "rec qnofz:=proc(phi::ratpoly,t::name,q::name,n::name,z::name)\nlocal gf,a; \n op(select(has,eval(subs(n=-n,a=subs(_A=q,proc(x) _A(-x-1) end), \n gfun[diffeqtorec](diff(gf(t),t)+(diff(phi,t)/z-diff(phi,t,t)/d iff(phi,t))*gf(t),gf(t),a(n)))),n))\nend:" "6#>%)recqnofzGR6''%$phiG%( ratpolyG'%\"tG%%nameG'%\"qGF,'%\"nGF,'%\"zGF,7$%#gfG%\"aG6\"F6-%#opG6# -%'selectG6%%$hasG-%%evalG6#-%%subsG6%/F0,$F0!\"\"/F5-FB6$/%#_AGF.R6#% \"xG7\"F6F6-FK6#,&FNFF\"\"\"FFF6F6F6-&%%gfunG6#%,diffeqtorecG6%,&-%%di ffG6$-F46#F+F+\"\"\"*&,&*&-Ffn6$F(F+FjnF2FFFjn*&-Ffn6%F(F+F+Fjn-Ffn6$F (F+FFFFFjn-F46#F+FjnFjn-F46#F+-F56#F0F0F6F6F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Example: " }{XPPEDIT 18 0 "phi = 1/((1-t)^2)-1;" "6#/%$ phiG,&*&\"\"\"\"\"\"*$,&\"\"\"F(%\"tG!\"\"\"\"#F-F(\"\"\"F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "qnRec:=recqnofz(1/(1-t)^2-1, t,q,n,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&qnRecG,**&,&*&%\"nG\" \"\"%\"zGF*!\"\"F+\"\"$F*-%\"qG6#,&F)F*F,F*F*F**&,&F+!\"*F(F-F*-F/6#,& F)F*!\"#F*F*F**&,(F(!\"$F+\"\"*F8F*F*-F/6#,&F)F*F;F*F*F**&,&F(F*F+F;F* -F/6#,&F)F*!\"%F*F*F*" }}}{EXCHG {PARA 274 "" 0 "" {TEXT -1 51 "We hav e obtained the same recurrence as when using " }{HYPERLNK 17 "Mgfun" 2 "Mgfun" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "qnRec:=applyop(factor,[2,2],applyop(collect,[2,1],readlib(isolate )(subs(n=n+1,qnRec),q(n)),q,normal));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&qnRecG/-%\"qG6#%\"nG,$*&,(*&,&%\"zG\"\"'*&F)\"\"\"F/F2!\"$F2-F'6 #,&F)F2!\"\"F2F2F2*&,(F/!\"'F1\"\"$\"\"#F2F2-F'6#,&F)F2!\"#F2F2F2*&,&F /F " 0 "" {MPLTEXT 1 0 27 "phi:=subs(z=u,solve(eq,f));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiG,$*&*&%\"uG\"\"\",&!\"#F)F(F)F)\"\"\"*$),&F(F)! \"\"F)\"\"#F,!\"\"F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "ass ume(z>0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "q0:=int(exp(-p hi/z)*diff(phi,u),u =0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q0G% #z|irG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "q0:=subs(z='z',q0 ); z:='z':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q0G%\"zG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Note that this initial value is a posteri ori obvious." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Thus we now have \+ both recurrence and initial condition. The solution " }{XPPEDIT 18 0 " q[n];" "6#&%\"qG6#%\"nG" }{TEXT -1 276 " to the recurrence equation qn Rec is a dominated solution, which means that any numerical error grow s exponentially. To avoid this, we run the recurrence backwards from a ny non trivial initial conditions. The dominating solution disappears \+ quickly, and we obtain the solution " }{XPPEDIT 18 0 "q[n];" "6#&%\"qG 6#%\"nG" }{TEXT -1 109 " because when the recurrence is run backwards \+ it becomes a dominating solution. We therefore add a parameter " } {XPPEDIT 18 0 "NN;" "6#%#NNG" }{TEXT -1 50 " indicating from where we \+ start running backwards." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "eval(collect(op(2,isolate(subs(n=n+3,qnRec),q(n))),q,normal),q=proc(n ) q(n,z,NN) end);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%\"qG6%,&%\"nG \"\"\"\"\"$F)%\"zG%#NNGF)-F%6%,&F(F)\"\"#F)F+F,!\"$*&*&,(F+F**&F(F)F+F )F*F0F)F)-F%6%,&F(F)F)F)F+F,F)\"\"\"*&F+\"\"\"F8\"\"\"!\"\"F)" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "qnprocrev:=subs(_REC=%,q=qnpro crev,proc(n,z,NN)\noption remember;\nif n=NN then 0 \nelif n=NN-1 then 0\nelif n=NN-2 then 1\nelse _REC fi end);" "6#>%*qnprocrevG-%%subsG6% /%%_RECG%\"%G/%\"qGF$R6%%\"nG%\"zG%#NNG7\"6#%)rememberG6\"@)/F/F1\"\"! /F/,&F1\"\"\"\"\"\"!\"\"F8/F/,&F1F;\"\"#F=\"\"\"F)F5F5F5" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%*qnprocrevGR6%%\"nG%\"zG%#NNG6\"6#%)rememberGF *@)/9$9&\"\"!/F/,&F0\"\"\"!\"\"F4F1/F/,&F0F4!\"#F4F4,(-F$6%,&F/F4\"\"$ F49%F0F4-F$6%,&F/F4\"\"#F4F>F0!\"$*&*&,(F>F=*&F/F4F>F4F=FBF4F4-F$6%,&F /F4F4F4F>F0F4\"\"\"*&F>\"\"\"FJ\"\"\"!\"\"F4F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Here is a procedure to compute " }{XPPEDIT 18 0 " q[n];" "6#&%\"qG6#%\"nG" }{TEXT -1 13 " numerically." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "`evalf/q/digits`:=0:" }}}{EXCHG {PARA 0 " > " 0 "" {XPPEDIT 19 1 "`evalf/q`:=proc (n,z,NN)\noption remember; \ng lobal `evalf/q`,`evalf/q/digits`;\nif Digits>`evalf/q/digits` \nthen ` evalf/q/digits`:=Digits;\n `evalf/q`:=subsop(4=NULL,op(`evalf/q`)) fi; \nif n = NN then 0 \n elif n = NN-1 then 0 \n elif n = NN-2 then \+ 1.0\n else \nprocname(n+3,z,NN)+(2+3*z+3*z*n)*procname(n+1,z,NN)/(z*(n +1))-3*procname(n+2,z,NN) fi end:" "6#>%(evalf/qGR6%%\"nG%\"zG%#NNG7\" 6#%)rememberG6\"C$@$2%/evalf/q/digitsG%'DigitsGC$>F1F2>F$-%'subsopG6$/ \"\"%%%NULLG-%#opG6#F$@)/F'F)\"\"!/F',&F)\"\"\"\"\"\"!\"\"FA/F',&F)FD \"\"#FF$\"#5!\"\",(-%)procnameG6%,&F'FD\"\"$FDF(F)FD*(,(\"\"#FD*&\"\"$ FDF(FDFD*(\"\"$FDF(FDF'FDFDFD-FO6%,&F'FD\"\"\"FDF(F)FD*&F(FD,&F'FD\"\" \"FDFDFFFD*&\"\"$FD-FO6%,&F'FD\"\"#FDF(F)FDFFF-6$F$F1F-" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "dnRec:=gfun[diffeqtorec](\{dneq,D(f)( 0)=1,f(0)=0\},f(z),a(n)):" "6#>%&dnRecG-&%%gfunG6#%,diffeqtorecG6%<%%% dneqG/--%\"DG6#%\"fG6#\"\"!\"\"\"/-F26#F4F4-F26#%\"zG-%\"aG6#%\"nG" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "dnproc:=gfun[rectoproc](dnR ec,a(n),remember):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Here is a p rocedure to compute " }{XPPEDIT 18 0 "d[n];" "6#&%\"dG6#%\"nG" }{TEXT -1 13 " numerically." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "`ev alf/d/digits`:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "`evalf/d `:=proc (n) option remember; global `evalf/d`,`evalf/d/digits`; if n>5 then if Digits>`evalf/d/digits` then `evalf/d/digits`:=Digits;`evalf/ d`:=subsop(4=NULL,op(`evalf/d`)) fi; evalf(-(-225*procname(n-5)-70*pro cname(n-1)+804*procname(n-4)-1011*procname(n-3)+514*procname(n-2)+(165 *procname(n-5)-693*procname(n-4)+1048*procname(n-3)-683*procname(n-2)+ 157*procname(n-1)+(-39*procname(n-5)+195*procname(n-4)-363*procname(n- 3)+309*procname(n-2)-114*procname(n-1)+(-18*procname(n-4)+42*procname( n-3)-48*procname(n-2)+3*procname(n-5)+27*procname(n-1))*n)*n)*n)/((-6+ (12-6*n)*n)*n)) elif n=0 then 0 elif n=1 then 1 elif n=2 then evalf(1/ 3) elif n=3 then evalf(-23/108) else evalf(-2749/3888) fi end:" "6#>%( evalf/dGR6#%\"nG7\"6#%)rememberG6\"@-2\"\"&F'C$@$2%/evalf/d/digitsG%'D igitsGC$>F2F3>F$-%'subsopG6$/\"\"%%%NULLG-%#opG6#F$-%&evalfG6#,$*&,.*& \"$D#\"\"\"-%)procnameG6#,&F'FH\"\"&!\"\"FHFN*&\"#qFH-FJ6#,&F'FH\"\"\" FNFHFN*&\"$/)FH-FJ6#,&F'FH\"\"%FNFHFH*&\"%65FH-FJ6#,&F'FH\"\"$FNFHFN*& \"$9&FH-FJ6#,&F'FH\"\"#FNFHFH*&,.*&\"$l\"FH-FJ6#,&F'FH\"\"&FNFHFH*&\"$ $pFH-FJ6#,&F'FH\"\"%FNFHFN*&\"%[5FH-FJ6#,&F'FH\"\"$FNFHFH*&\"$$oFH-FJ6 #,&F'FH\"\"#FNFHFN*&\"$d\"FH-FJ6#,&F'FH\"\"\"FNFHFH*&,.*&\"#RFH-FJ6#,& F'FH\"\"&FNFHFN*&\"$&>FH-FJ6#,&F'FH\"\"%FNFHFH*&\"$j$FH-FJ6#,&F'FH\"\" $FNFHFN*&\"$4$FH-FJ6#,&F'FH\"\"#FNFHFH*&\"$9\"FH-FJ6#,&F'FH\"\"\"FNFHF N*&,,*&\"#=FH-FJ6#,&F'FH\"\"%FNFHFN*&\"#UFH-FJ6#,&F'FH\"\"$FNFHFH*&\"# [FH-FJ6#,&F'FH\"\"#FNFHFN*&\"\"$FH-FJ6#,&F'FH\"\"&FNFHFH*&\"#FFH-FJ6#, &F'FH\"\"\"FNFHFHFHF'FHFHFHF'FHFHFHF'FHFHFH*&,&\"\"'FN*&,&\"#7FH*&\"\" 'FHF'FHFNFHF'FHFHFHF'FHFNFN/F'\"\"!Fju/F'\"\"\"\"\"\"/F'\"\"#-FA6#*&\" \"\"FH\"\"$FN/F'\"\"$-FA6#,$*&\"#BFH\"$3\"FNFN-FA6#,$*&\"%\\FFH\"%))QF NFNF+6$F$F2F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "Finally, the fo llowing procedure computes values of the double confluent Heun functio n as follows. First, an upper value of " }{XPPEDIT 18 0 "NN;" "6#%#NNG " }{TEXT -1 17 " is selected and " }{XPPEDIT 18 0 "2*NN;" "6#*&\"\"#\" \"\"%#NNGF%" }{TEXT -1 70 " is used in the backward recurrence to comp ute the scaling to use for " }{XPPEDIT 18 0 "q[n](z);" "6#-&%\"qG6#%\" nG6#%\"zG" }{TEXT -1 84 " in view of the actual initial condition. The n the summation is performed up to the " }{XPPEDIT 18 0 "NN;" "6#%#NNG " }{TEXT -1 63 "th term. If the relative error of the last term is lar ger than " }{XPPEDIT 18 0 "10^(-Digits);" "6#)\"#5,$%'DigitsG!\"\"" } {TEXT -1 7 ", then " }{XPPEDIT 18 0 "NN;" "6#%#NNG" }{TEXT -1 89 " is \+ doubled and the computation starts again. Note that option remember ha s been used in " }{XPPEDIT 18 0 "qnprocrev;" "6#%*qnprocrevG" }{TEXT -1 45 " so as to avoid duplicating some of the work." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "time(evalf(q(0,10.2,3000),21));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$*R!\"$" }}}{EXCHG {PARA 0 "> " 0 ""  {MPLTEXT 1 0 450 "valheun:=subs(_q0=q0,proc(z)\n local tot,i,N,NN,l ambda,st,D;\nN:=10;\nst:=time();\ndo \n N:=floor(2*N);\n NN:=N+f loor(sqrt(N))+10;\n D:=Digits+3*ilog10(N) +floor(log(N));\n lamb da:=_q0/evalf(q(0,z,NN),D);\n \n tot:=add(evalf(d(i),D)*evalf(q(i ,z,NN),D),i=1..N)*lambda;\n if abs(evalf(d(N),D)*evalf(q(N,z,NN),D) *lambda) " 0 "" {XPPEDIT 19 1 "infolevel[ valheun]:=1:" "6#>&%*infolevelG6#%(valheunG\"\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 20 "plot(valheun,0..50);" }}{PARA 6 "" 1 "" {TEXT -1 75 "valheun: \"N=\" 80 \"z=\" 1.089857709 \"time:\" .141 \"digits:\" 17" }}{PARA 6 "" 1 "" {TEXT -1 76 "valheun: \+ \"N=\" 160 \"z=\" 2.038137074 \"time:\" .270 \"digits:\" \+ 21" }}{PARA 6 "" 1 "" {TEXT -1 76 "valheun: \"N=\" 320 \"z=\" \+ 3.104576397 \"time:\" .579 \"digits:\" 21" }}{PARA 6 "" 1 "" {TEXT -1 76 "valheun: \"N=\" 320 \"z=\" 4.178084772 \"time: \" .571 \"digits:\" 21" }}{PARA 6 "" 1 "" {TEXT -1 76 "valheun: \+ \"N=\" 320 \"z=\" 5.246490950 \"time:\" .599 \"digits:\" 21" }}{PARA 6 "" 1 "" {TEXT -1 77 "valheun: \"N=\" 640 \"z=\" 6.237040242 \"time:\" 1.221 \"digits:\" 22" }}{PARA 6 "" 1 "" {TEXT -1 77 "valheun: \"N=\" 640 \"z=\" 7.262696659 \"tim e:\" 1.080 \"digits:\" 22" }}{PARA 6 "" 1 "" {TEXT -1 77 "valheu n: \"N=\" 640 \"z=\" 8.323431992 \"time:\" 1.129 \"digit s:\" 22" }}{PARA 6 "" 1 "" {TEXT -1 77 "valheun: \"N=\" 640 \" z=\" 9.380765534 \"time:\" 1.240 \"digits:\" 22" }}{PARA 6 " " 1 "" {TEXT -1 77 "valheun: \"N=\" 640 \"z=\" 10.46836305 \+ \"time:\" 1.201 \"digits:\" 22" }}{PARA 6 "" 1 "" {TEXT -1 78 "v alheun: \"N=\" 1280 \"z=\" 11.42631952 \"time:\" 2.520 \+ \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 1 280 \"z=\" 12.50475163 \"time:\" 2.369 \"digits:\" 26" }} {PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 1280 \"z=\" 13.58 761188 \"time:\" 2.601 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 1280 \"z=\" 14.63114838 \"time: \" 2.750 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 1280 \"z=\" 15.57877949 \"time:\" 2.800 \"digits :\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 1280 \" z=\" 16.70560455 \"time:\" 3.110 \"digits:\" 26" }}{PARA 6 " " 1 "" {TEXT -1 78 "valheun: \"N=\" 1280 \"z=\" 17.66017317 \+ \"time:\" 3.219 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "v alheun: \"N=\" 1280 \"z=\" 18.77056341 \"time:\" 3.340 \+ \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 2 560  \"z=\" 19.75344692 \"time:\" 7.290 \"digits:\" 26" }} {PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 2560 \"z=\" 20.83 182591 \"time:\" 8.091 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 2560 \"z=\" 21.85869773 \"time: \" 8.440 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 2560 \"z=\" 22.93013074 \"time:\" 9.180 \"digits :\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 2560 \" z=\" 23.91404071 \"time:\" 9.710 \"digits:\" 26" }}{PARA 6 " " 1 "" {TEXT -1 79 "valheun: \"N=\" 2560 \"z=\" 24.97532053 \+ \"time:\" 10.209 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 79 " valheun: \"N=\" 2560 \"z=\" 26.07769200 \"time:\" 10.941 \+ \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" \+ 2560 \"z=\" 27.03730960 \"time:\" 11.229 \"digits:\" 26" } }{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 2560 \"z=\" 28.0 7372290 \"time:\" 11.961  \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 2560 \"z=\" 29.14443926 \"time: \" 12.589 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun : \"N=\" 2560 \"z=\" 30.19192632 \"time:\" 12.810 \"digi ts:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 2560 \+ \"z=\" 31.20542392 \"time:\" 13.741 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 32.33074020 \"time:\" 20.060 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 33.34188693 \"time:\" 12.8 80 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N= \" 5120 \"z=\" 34.42150189 \"time:\" 15.039 \"digits:\" \+ 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" \+ 35.39979350 \"time:\" 17.241 \"digits:\" 27" }}{PARA 6 "" 1 " " {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 36.46932464 \"tim e:\" 19.520 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valhe un: \"N=\" 5120 \"z=\" 37.47567008 \"time:\" 21.760 \"di gits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \+ \"z=\" 38.52759104 \"time:\" 24.110 \"digits:\" 27" }} {PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 39.55 603605 \"time:\" 26.450 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 40.63272267 \"time: \" 28.740 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun : \"N=\" 5120 \"z=\" 41.66969985 \"time:\" 31.479 \"digi ts:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \+ \"z=\" 42.73015878 \"time:\" 33.841 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 43.78183659 \"time:\" 36.099 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 44.74821926 \"time:\" 39.0 00 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N= \" 5120 \"z=\" 45.85580287 \"time:\" 40.981 \"digits:\" \+ 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" \+ 46.84643885 \"time:\" 43.800 \"digits:\" 27" }}{PARA 6 "" 1 " " {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 47.90266342 \"tim e:\" 45.850 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valhe un: \"N=\" 5120 \"z=\" 48.91360511 \"time:\" 48.260 \"di gits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 72 "valheun: \"N=\" 5120 \+ \"z=\" 50.0 \"time:\" 50.699 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 77 "valheun: \"N=\" 160 \"z=\" 1.563997392 \"t ime:\" 1.450 \"digits:\" 21" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'C URVESG6$7S7$$\"+4x&)*3\"!\"*$\"+7+&3\"H!#57$$\"+#R(*Rc\"F*$\"+mL$Q6&F- 7$$\"+uq8Q?F*$\"+&Q!)ps(F-7$$\"+(RwX5$F*$\"+'epd]\"F*7$$\"+sZ3yTF*$\"+ $Q+sV#F*7$$\"+]4\\Y_F*$\"+gn*z`$F*7$$\"+U-/PiF*$\"+w!)*[p%F*7$$\"+fmpi sF*$\"+nI,9gF*7$$\"+#*>VB$)F*$\"+\"=:D\\(F*7$$\"+Mbw!Q*F*$\"+**)\\'o!* F*7$$\"+0j$o/\"!\")$\"+CI\\y5Fhn7$$\"+_>jU6Fhn$\"+N4'oB\"Fhn7$$\"+j^Z] 7Fhn$\"+4#pCU\"Fhn7$$\"+)=h(e8Fhn$\"+FQ(fh\"Fhn7$$\"+Q[6j9Fhn$\"+1_l3= Fhn7$$\"+\\z(yb\"Fhn$\"+ny]))>Fhn7$$\"+b/cq;Fhn$\"+Y8)z?#Fhn7$$\"+Fhn$\"+4W#*GGF hn7$$\"+\"f#=$3#Fhn$\"+-(*>dIFhn7$$\"+t(pe=#Fhn$\"+.OMyKFhn7$$\"+uI,$H #Fhn$\"+`+z7NFhn7$$\"+rSS\"R#Fhn$\"+enCJPFhn7$$\"+`?`(\\#Fhn$\"+\"30,( RFhn7$$\"++#pxg#Fhn$\"+J7e@UFhn7$$\"+g4t.FFhn$\"+&\\cJW%Fhn7$$\"+!Hst! GFhn$\"+RW7&o%Fhn7$$\"+ERW9HFhn$\"+.%yy$\\Fhn7$$\"+KE>>IFhn$\"+wKw(=&F hn7$$\"+#RU07$Fhn$\"+;#**=V&Fhn7$$\"+?S2LKFhn$\"+Q%zbq&Fhn7$$\"+$p)=ML Fhn$\"+(HVP&fFhn7$$\"+*=]@W$Fhn$\"+:3(4A'Fhn7$$\"+]$z*RNFhn$\"+j?1lkFh n7$$\"+kC$pk$Fhn$\"+df&Rt'Fhn7$$\"+3qcZPFhn$\"+uH%)))pFhn7$$\"+/\"fF&Q Fhn$\"+sQ:dsFhn7$$\"+0OgbRFhn$\"+DKF@vFhn7$$\"+nAFjSFhn$\"+Cdi*z(Fhn7$ $\"+&)*pp;%Fhn$\"+>yVp!)Fhn7$$\"+ye,tUFhn$\"+G40Z$)Fhn7$$\"+fO=yVFhn$ \"+Z!3Si)Fhn7$$\"+E>#[Z%Fhn$\"+$Q-*z))Fhn7$$\"+(G!e&e%Fhn$\"+Wnyu\"*Fh n7$$\"+&)Qk%o%Fhn$\"+`)Q*R%*Fhn7$$\"+UjE!z%Fhn$\"+:%ySs*Fhn7$$\"+60O\" *[Fhn$\"+87Q(***Fhn7$\"#]$\"+U%[#H5!\"(-%'COLOURG6&%$RGBG$\"#5!\"\"\" \"!F`[l-%+AXESLABELSG6$%!GFd[l-%%VIEWG6$;F`[l$FezF`[l%(DEFAULTG" 2 762 762 762 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 2 0 0 0 0 0 0 }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 117 "This curve is to be contrasted with the irregular plot we got from the same series using summation to the least term." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 216 "Sections 1 and 2 of this works heet only depend on the Heun differential equation, and can easily be \+ adapted to any linear differential equation. Sections 3 and 4 compute \+ the recurrences satisfied by the coefficients " }{XPPEDIT 18 0 "q[n]; " "6#&%\"qG6#%\"nG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "d[n];" "6#&%\" dG6#%\"nG" }{TEXT -1 53 ", which depend on the choice of the mapping f unction " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 10 " only for " } {XPPEDIT 18 0 "q[n];" "6#&%\"qG6#%\"nG" }{TEXT -1 47 ", and on the dif ferential equation as well for " }{XPPEDIT 18 0 "d[n];" "6#&%\"dG6#%\" nG" }{TEXT -1 62 ". Section 5 details the numerical computations. It d epends on " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 158 " and on th e recurrences found in the previous sections. This worksheet can be ad apted to another mapping function and to another linear differential e quation." }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 10 "References" }}{PARA 0 "" 0 "" {TEXT -1 15 "[Lutz et al.] " }{TEXT 268 40 "On the converge nce of Borel approximants" }{TEXT -1 62 ", by W. Balser, D. A. Lutz a nd R. Sch\344fke, (2000). Preprint." }}{PARA 0 "" 0 "" {TEXT -1 12 "[ DuLoRi92] " }{TEXT 269 77 "Kovacic's Algorithm and Its Application to Some Families of Special Functions" }{TEXT -1 43 ", by Anne Duval and Mich\350le Loday-Richaud, " }{TEXT 270 62 "Applicable Algebra in Engi neering, Communication and Computing" }{TEXT -1 29 ", (1992), vol. 3, \+ p. 211-246." }}{PARA 0 "" 0 "" {TEXT -1 10 "[Durand] " }{TEXT 272 75 "On the convergence of Borel approximants [summary of a talk by Donald Lutz]" }{TEXT -1 45 ", by Marianne Durand, (2001). To appear in: " } {TEXT 271 29 "Algorithms Seminar, 2000-2001" }{TEXT -1 24 ", INRIA Res earch Report." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 } ctions. This worksheet can be ad apted to another mapping function and to another linear differential e quation." }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 10 "References" }}{PARA 0 "" 0 "" {TEXT -1 15 "[Lutz et al.] " }{TEXT 268 40 "On the converge nce of Borel approximants" }{TEXT -1 62 ", by W. Balser, D. A. Lutz a nd R. Sch\344fke, (2000). Preprint." }}{PARA 0 "" 0 "" {TEXT -1 12 "[ DuLoRi92] " }{TEXT 269 77 "Kovacic's Algorithm and Its Application to Some Families of Special Functions" }{TEXT -1 43 ", by Anne Duval and Mich\350le Loday-Richaud, " }{TEXT 270 62 "Applicable Algebra in Engi neering, Communication and Computing" }{TEXT -1 29 ", (1992), vol. 3, \+ p. 211-246." }}{PARA 0 "" 0 "" {TEXT -1 10 "[Durand] " }{TEXT 272 75 "On the convergence of Borel approximants 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2 3 "DEC ALPHA UNIX" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "2 D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 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0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 16 "GFUN AND THE AGM" }} {PARA 19 "" 0 "" {TEXT 257 11 "Bruno Salvy" }}{PARA 261 "" 0 "" {TEXT -1 12 "January 1998" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 4 "Let " }{XPPEDIT 18 0 "a" "I\"aG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b" "I\"bG6\"" }{TEXT -1 36 " be two positive real numb ers, with " }{XPPEDIT 18 0 "a>b" "2%\"bG%\"aG" }{TEXT -1 6 ". The " } {TEXT 258 20 "arithmetic-geometric" }{TEXT -1 9 " mean of " }{XPPEDIT 18 0 "a" "I\"aG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b" "I\"bG6\"" }{TEXT -1 61 " is classically defined as the common limit of the seque nces " }{XPPEDIT 18 0 "a[k]" "&%\"aG6#%\"kG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "b[k]" "&%\"bG6#%\"kG" }{TEXT -1 12 " defined by\n" } {XPPEDIT 18 0 "a[k+1]=(a[k]+b[k])/2, b[k+1]=sqrt(a[k]*b[k])" "6$/&%\"a G6#,&%\"kG\"\"\"\"\"\"F)*&,&&F%6#F(F)&%\"bG6#F(F)F)\"\"#!\"\"/&F06#,&F (F)\"\"\"F)-%%sqrtG6#*&&F%6#F(F)&F06#F(F)" }{TEXT -1 7 ", with " } {XPPEDIT 18 0 "a[0]=a" "/&%\"aG6#\"\"!F$" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "b[0]=b" "/&%\"bG6#\"\"!F$" }{TEXT -1 68 ".\nThat the se quences converge to the same limit can be inferred from" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k+1]^2-b[k+1]^2=((a [k]-b[k])/2)^2" "/,&*$&%\"aG6#,&%\"kG\"\"\"\"\"\"F*\"\"#F**$&%\"bG6#,& F)F*\"\"\"F*\"\"#!\"\"*$*&,&&F&6#F)F*&F/6#F)F4F*\"\"#F4\"\"#" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "This common limit is k nown by Maple as " }{HYPERLNK 17 "GaussAGM" 2 "GaussAGM" "" }{XPPEDIT 18 0 "``(a,b)" "-%!G6$%\"aG%\"bG" }{TEXT -1 105 ". It was discovered b y Gauss that the arithmetic-geometric mean is related to hypergeometri c functions by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "GaussAGM( a,b)=a/hypergeom([1/2, 1/2],[1],1-b^2/a^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)GaussAGMG6$%\"aG%\"bG*&F'\"\"\"-%*hypergeomG6%7$#F* \"\"#F/7#F*,&F*F**&F(F0F'!\"#!\"\"F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eval(subs(a=3.,b=2.,\"));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"+O/ouC!\"*$\"+P/ouCF&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "This worksheet, largely inspired by [1], shows how " } {HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 123 " can be used to guess a nd then prove this result, as well as a generalization of it due to J. M. Borwein and P. B. Borwein." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 23 "The functional equation" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 143 "Fo llowing [1], we start by introducing a generalization of the arithmeti c-geometric mean obtained by considering the following iteration where " }{XPPEDIT 18 0 "N>1" "2\"\"\"%\"NG" }{TEXT -1 16 " is an integer: \+ " }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k+1]=(a [k]+(N-1)*b[k])/N,b[k+1]=(a[k+1]^N-((a[k]-b[k])/N)^N)^(1/N)" "6$/&%\"a G6#,&%\"kG\"\"\"\"\"\"F)*&,&&F%6#F(F)*&,&%\"NGF)\"\"\"!\"\"F)&%\"bG6#F (F)F)F)F1F3/&F56#,&F(F)\"\"\"F)),&)&F%6#,&F(F)\"\"\"F)F1F))*&,&&F%6#F( F)&F56#F(F3F)F1F3F1F3*&\"\"\"F)F1F3" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "where the second equality is motivated by" }}} {EXCHG {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k+1]^N-b[k+1 ]^N=((a[k]-b[k])/N)^N" "/,&)&%\"aG6#,&%\"kG\"\"\"\"\"\"F*%\"NGF*)&%\"b G6#,&F)F*\"\"\"F*F,!\"\")*&,&&F&6#F)F*&F/6#F)F3F*F,F3F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "from which follows that both sequences co nverge to a common limit, which is denoted by " }{XPPEDIT 18 0 "M[N](a ,b)" "-&%\"MG6#%\"NG6$%\"aG%\"bG" }{TEXT -1 56 ". The arithmetic-geome tric mean corresponds to the case " }{XPPEDIT 18 0 "N=2" "/%\"NG\"\"# " }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The function \+ " }{XPPEDIT 18 0 "M[N](a,b)" "-&%\"MG6#%\"NG6$%\"aG%\"bG" }{TEXT -1 35 " is easily seen to be homogeneous: " }{XPPEDIT 18 0 "M[N](lambda*a ,lambda*b)=lambda*M[N](a,b)" "/-&%\"MG6#%\"NG6$*&%'lambdaG\"\"\"%\"aGF +*&F*F+%\"bGF+*&F*F+-&F%6#F'6$F,F.F+" }{TEXT -1 6 ", for " }{XPPEDIT 18 0 "lambda>0" "2\"\"!%'lambdaG" }{TEXT -1 42 ". Together with the ob vious property that " }{XPPEDIT 18 0 "M[N](a[0],b[0])=M[N](a[1],b[1]) " "/-&%\"MG6#%\"NG6$&%\"aG6#\"\"!&%\"bG6#F,-&F%6#F'6$&F*6#\"\"\"&F.6# \"\"\"" }{TEXT -1 24 ", this implies that for " }{XPPEDIT 18 0 "x" "I \"xG6\"" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "` `(0,1)" "-%\"~G6$\"\"!\" \"\"" }{TEXT -1 1 "," }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "M[N](1,(1-x^N)^(1/N))=(1+(N-1)*x)*M[N](1,(1-x)/(1+(N-1) *x))" "/-&%\"MG6#%\"NG6$\"\"\"),&\"\"\"\"\"\")%\"xGF'!\"\"*&\"\"\"F-F' F0*&,&\"\"\"F-*&,&F'F-\"\"\"F0F-F/F-F-F--&F%6#F'6$\"\"\"*&,&\"\"\"F-F/ F0F-,&\"\"\"F-*&,&F'F-\"\"\"F0F-F/F-F-F0F-" }{TEXT -1 2 ". " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Defining the function " }{XPPEDIT 18 0 "A[N](x)" "-&%\"AG6#%\"NG6#%\"xG" }{TEXT -1 3 " by" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[N](x)=1/M[N](1,(1-x )^(1/N))" "/-&%\"AG6#%\"NG6#%\"xG*&\"\"\"\"\"\"-&%\"MG6#F'6$\"\"\"),& \"\"\"F,F)!\"\"*&\"\"\"F,F'F6F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "the equation above translates into the following " }{TEXT 259 19 " functional equation" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "A[N](x)" "-&% \"AG6#%\"NG6#%\"xG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "funeq:=(1+(N-1)*x)*A[N](x^N)=A[N](1-((1-x)/(1+(N-1)*x)) ^N):" ">%&funeqG/*&,&\"\"\"\"\"\"*&,&%\"NGF(\"\"\"!\"\"F(%\"xGF(F(F(-& %\"AG6#F+6#)F.F+F(-&F16#F+6#,&\"\"\"F()*&,&\"\"\"F(F.F-F(,&\"\"\"F(*&, &F+F(\"\"\"F-F(F.F(F(F-F+F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "wh ich plays a central r\364le in this worksheet. It is not too difficult to show that " }{XPPEDIT 18 0 "A[N](x)" "-&%\"AG6#%\"NG6#%\"xG" } {TEXT -1 139 " is analytic in the neighborhood of the origin and that \+ the functional equation above has a unique analytic solution in this n eighborhood. " }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "The quadratic \+ case" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "This is the case " } {XPPEDIT 18 0 "N=2" "/%\"NG\"\"#" }{TEXT -1 51 " and Gauss's theorem i s equivalent to stating that " }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[2](x)=hypergeom([1/2,1/2],[1],x)" "/-&%\"AG6#\" \"#6#%\"xG-%*hypergeomG6%7$*&\"\"\"\"\"\"\"\"#!\"\"*&\"\"\"F0\"\"#F27# \"\"\"F)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "We no w use " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 126 " to first gues s and then prove this result. The first step is to use the functional \+ equation to compute a series expansion of " }{XPPEDIT 18 0 "A[2](x)" " -&%\"AG6#\"\"#6#%\"xG" }{TEXT -1 168 ", then we use this series to gue ss a possible closed form which turns out to be analytic, then we show that this analytic function does satisfy the functional equation." }} }{SECT 0 {PARA 4 "" 0 "" {TEXT -1 16 "Series expansion" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Starting from the functional equation," } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "funeq2:=subs(N=2,A[2]=A,op (1,funeq)-op(2,funeq));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'funeq2G, &*&,&\"\"\"F(%\"xGF(F(-%\"AG6#*$F)\"\"#F(F(-F+6#,&F(F(*&,&F(F(F)!\"\"F .F'!\"#F4F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "a series solution \+ is easily obtained by a method of undeterminate coefficients:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "sol:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "for i to 12 do \n sol:=sol+x^i*solve(op(1, series(eval(\n subs(A=unapply(sol+a*x^i,x),funeq2)),x,i+2)),a) od:so l;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,<\"\"\"F$%\"xG#F$\"\"%*$F%\"\"# #\"\"*\"#k*$F%\"\"$#\"#D\"$c#*$F%F'#\"%D7\"&%Q;*$F%\"\"&#\"%pR\"&Ob'*$ F%\"\"'#\"&hL&\"(w&[5*$F%\"\"(#\"'TS=\"(/V>%*$F%\"\")#\")D#49%\"+C=ut5 *$F%F+#\"*DSuZ\"\"+'Hn\\H%*$F%\"#5#\"+@PUL@\",OnZ>(o*$F%\"#6#\"+Tg`vx \"-Wp!z([F*$F%\"#7#\"-@&H(GqX\"/;W/'=#f<" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 "Guessing the solution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "From this series, " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 61 " guesses a differential equation which could be satisfied by " } {XPPEDIT 18 0 "A(x)" "-%\"AG6#%\"xG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "deq:=op(1,gfun[seriestodiffeq](series(sol ,x,13),y(x),[ogf]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$deqG<%/-%\" yG6#\"\"!\"\"\",(-F(6#%\"xGF+*&,&!\"%F+F/\"\")F+-%%diffG6$F-F/F+F+*&,& F/F2*$F/\"\"#\"\"%F+-F56$F4F/F+F+/--%\"DG6#F(F)#F+F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "It turns out that Maple's dsolve function is un able to solve this differential equation:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "dsolve(deq,y(x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "We then use " }{HYPERLNK 17 "gfun[diffeqtorec]" 2 "gfun[diffeqt orec]" "" }{TEXT -1 116 " which deduces from this differential equatio n the recurrence satisfied by the Taylor coefficients of its solutions :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "gfun[diffeqtorec](deq, y(x),u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%,&*&,(\"\"\"F'%\"nG\" \"%*$F(\"\"#F)F'-%\"uG6#F(F'F'*&,(F(!\")!\"%F'F*F2F'-F-6#,&F(F'F'F'F'F '/-F-6#\"\"!F'/-F-6#F'#F'F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Fr om this first order linear recurrence, a solution is easily found:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "rsolve(\",u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%&GAMMAG6#,&%\"nG\"\"\"#F)\"\"#F)F+-F%6#, &F(F)F)F)!\"#%#PiG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "hence \+ the sum:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "y(x)=sum(\"*x^n ,n=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%*h ypergeomG6%7$#\"\"\"\"\"#F,7#F-F'" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 31 "Proving the result of the guess" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "The proof consists in showing that the function " } {XPPEDIT 18 0 "y(x)" "-%\"yG6#%\"xG" }{TEXT -1 64 ", which is obviousl y analytic, satisfies the functional equation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(A=y,funeq2)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,&\"\"\"F'%\"xGF'F'-%\"yG6#*$F(\"\"#F'F'-F*6#,&F'F '*&,&F'F'F(!\"\"F-F&!\"#F3F3\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "Our approach consists in using closure properties of solutions of linear differential equations that are implemented in " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 284 " to compute a linear differential equation satisfied by the left-hand side of this equation. The proof \+ then reduces to showing that 0 is the only solution of this differenti al equation that is compatible with the initial conditions, which are \+ 0 up to a large order by construction of " }{XPPEDIT 18 0 "y" "I\"yG6 \"" }{TEXT -1 144 ".\nIt turns out that this proof can be performed di rectly from the differential equation, and would apply even if no clos ed-form had been found. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Given a linear differential equation satisfied by a series " }{XPPEDIT 18 0 "y(x)" "-%\"yG6#%\"xG" }{TEXT -1 15 ", the function " }{HYPERLNK 17 "gfun[algebraicsubs]" 2 "gfun[algebraicsubs]" "" }{TEXT -1 55 " comput es a linear differential equation satisfied by " }{XPPEDIT 18 0 "y(f( x))" "-%\"yG6#-%\"fG6#%\"xG" }{TEXT -1 51 " for any algebraic function , given by a polynomial " }{XPPEDIT 18 0 "P" "I\"PG6\"" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "P(x,f(x))=0" "/-%\"PG6$%\"xG-%\"fG6#F&\" \"!" }{TEXT -1 44 ". Thus a differential equation satisfied by " } {XPPEDIT 18 0 "y(1-(1-x)^2/(1+x)^2)" "-%\"yG6#,&\"\"\"\"\"\"*&,&\"\"\" F'%\"xG!\"\"\"\"#*$,&\"\"\"F'F+F'\"\"#F,F," }{TEXT -1 43 " is easily c omputed from that satisfied by " }{XPPEDIT 18 0 "y(x)" "-%\"yG6#%\"xG " }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "deq:=op (select(has,deq,x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "deq 1:=gfun[algebraicsubs](deq,numer(y-(1-(1-x)^2/(1+x)^2)),y(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%deq1G,(*&,&!\"\"\"\"\"%\"xGF)F)-%\" yG6#F*F)F)*&,**$F*\"\"$F(*$F*\"\"#!\"$F*F(F)F)F)-%%diffG6$F+F*F)F)*&,* *$F*\"\"%F(F0F(F2F)F*F)F)-F66$F5F*F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Similarly, " }{XPPEDIT 18 0 "y(x^2)" "-%\"yG6#*$%\"xG\"\" #" }{TEXT -1 10 " satisfies" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "gfun[algebraicsubs](deq,y-x^2,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&-%\"yG6#%\"xG\"\"\"F(F)F)*&,&!\"\"F)*$F(\"\"#\"\"$F)-%%diffG 6$F%F(F)F)*&,&F(F,*$F(F/F)F)-F16$F0F(F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "and its product by " }{XPPEDIT 18 0 "-(1+x)" ",$,&\"\"\" \"\"\"%\"xGF%!\"\"" }{TEXT -1 10 " satisfies" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "deq2:=gfun[`diffeq*diffeq`](\",y(x)+1+x,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%deq2G,(*&,&\"\"\"F(%\"xG!\"\"F(-% \"yG6#F)F(F(*&,*F)F(*$F)\"\"$F(*$F)\"\"#F1F*F(F(-%%diffG6$F+F)F(F(*&,* F)F*F2F*F0F(*$F)\"\"%F(F(-F56$F4F)F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 165 "From there, we deduce a differential equation satisfied \+ by the left-hand side of the functional equation when applied to the h ypergeometric function we have guessed:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "gfun[`diffeq+diffeq`](deq1,deq2,y(x));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,(*&,&\"\"\"F&%\"xG!\"\"F&-%\"yG6#F'F&F&*&,*F'F& *$F'\"\"$F&*$F'\"\"#F/F(F&F&-%%diffG6$F)F'F&F&*&,*F'F(F0F(F.F&*$F'\"\" %F&F&-F36$F2F'F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Analytic so lutions of this equation have a coefficient sequence which satisfies \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "gfun[diffeqtorec](\",y( x),u(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<&,**&%\"nG\"\"#-%\"uG6#F &\"\"\"F+*&,(F'F+*$F&F'F+F&\"\"%F+-F)6#,&F&F+F+F+F+F+*&,(F&!\"#F+F+F.! \"\"F+-F)6#,&F&F+F'F+F+F+*&,(F&!\"'!\"*F+F.F6F+-F)6#,&F&F+\"\"$F+F+F+/ -F)6#F+,$&%#_CG6#\"\"!F//-F)6#F'FF/-F)FHFE" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "and thus the first three zeroes of the Taylor expansion \+ of the left-hand side of the functional equation conclude the proof." }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "The cubic case" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "It has been discovered by J. M. Borwein \+ and P. B. Borwein that a hypergeometric expression also exists when " }{XPPEDIT 18 0 "N=3" "/%\"NG\"\"$" }{TEXT -1 91 ". Again, the same ste ps as above lead to guessing and then proving the following result by \+ " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 " " 0 "" {TEXT 262 7 "Theorem" }{TEXT -1 25 ". [Borwein & Borwein 90] " }{TEXT 263 13 "The function " }{XPPEDIT 264 0 "A[3](x)" "-&%\"AG6#\"\" $6#%\"xG" }{TEXT 265 77 " corresponding to the AGM iteration of order \+ 3 has the following closed form:" }}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[3](x)=hypergeom([1/3,2/3],[1],x)" "/-&%\"AG 6#\"\"$6#%\"xG-%*hypergeomG6%7$*&\"\"\"\"\"\"\"\"$!\"\"*&\"\"#F0\"\"$F 27#\"\"\"F)" }{TEXT -1 1 "." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 16 "S eries expansion" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "We start from t he functional equation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "f uneq3:=subs(N=3,A[3]=A,op(1,funeq)-op(2,funeq));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'funeq3G,&*&,&\"\"\"F(%\"xG\"\"#F(-%\"AG6#*$F)\"\"$F( F(-F,6#,&F(F(*&,&F(F(F)!\"\"F/F'!\"$F5F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "and compute the first terms of the series expansion of th e solution:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "sol:=1:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "for i to 12 do \n sol:=so l+x^i*solve(op(1,series(eval(\n subs(A=unapply(sol+a*x^i,x),funeq3)) ,x,i+2)),a) od:sol;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,<\"\"\"F$%\"xG #\"\"#\"\"**$F%F'#\"#5\"#\")*$F%\"\"$#\"$g&\"%hl*$F%\"\"%#\"%]Q\"&\\!f *$F%\"\"&#\"&G!G\"'T9`*$F%\"\"'#\"(/f!>\")@n/V*$F%\"\"(#\")![!y9\"**[? uQ*$F%\"\")#\"*q\"eo6\"+,Wy'[$*$F%F(#\",+0\"y&f(\".H$Ge'=a#*$F%F+#\"-g 7Uxnh\"/h\\X#zwG#*$F%\"#6#\".?vQbl0&\"0\\Y4K6*e?*$F%\"#7#\"0+o%pI\"3w$ \"2plm*p\"=xm\"" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 "Guessing the solution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Again, this is a luck y situation where a differential equation can be guessed:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "deq:=op(1,gfun[seriestodiffeq](seri es(sol,x,13),y(x),[ogf]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$deqG< %/--%\"DG6#%\"yG6#\"\"!#\"\"#\"\"*/-F+F,\"\"\",(-F+6#%\"xGF/*&,&!\"*F3 F7\"#=F3-%%diffG6$F5F7F3F3*&,&F7F:*$F7F/F0F3-F=6$F " 0 "" {MPLTEXT 1 0 33 "gfun[diffeqtorec](deq ,y(x),u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%\"uG6#\"\"\"#\"\" #\"\"*/-F&6#\"\"!F(,&*&,(F*F(%\"nGF+*$F3F*F+F(-F&6#F3F(F(*&,(F3!#=!\"* F(F4F:F(-F&6#,&F3F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "rsolve(\",u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,-%&GAMMAG6 #,&%\"nG\"\"\"#\"\"#\"\"$F*F*-F&6#,&F)F*#F*F-F*F*-F&6#,&F)F*F*F*!\"#%# PiG!\"\"F-#F*F,F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "y(x)=s um(\"*x^n,n=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6# %\"xG-%*hypergeomG6%7$#\"\"\"\"\"$#\"\"#F.7#F-F'" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 31 "Proving the result of the guess" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "The same routine applies:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(A=y,funeq3)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,&\"\"\"F'%\"xG\"\"#F'-%\"yG6#*$F(\"\"$F'F'-F+6#,& F'F'*&,&F'F'F(!\"\"F.F&!\"$F4F4\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "deq:=op(select(has,deq,x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "deq1:=gfun[algebraicsubs](deq,numer(y-(1-(1-x)^3 /(1+2*x)^3)),y(x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%deq1G,(*&,( \"\"#\"\"\"%\"xG!\"%*$F*F(F(F)-%\"yG6#F*F)F)*&,,!\"\"F)*$F*\"\"&\"\")* $F*\"\"%\"#7*$F*\"\"$F7F,F7F)-%%diffG6$F-F*F)F)*&,.*$F*\"\"'F7F3F7F6F) F9F+F,F+F*F2F)-F<6$F;F*F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "gfun[algebraicsubs](deq,y-x^3,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&-%\"yG6#%\"xG\"\"\"F(\"\"#F**&,&!\"\"F)*$F(\"\"$\"\"%F)-%%di ffG6$F%F(F)F)*&,&F(F-*$F(F0F)F)-F26$F1F(F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "deq2:=gfun[`diffeq*diffeq`](\",y(x)+1+2*x,y(x)); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%deq2G,(*&,(\"\"#\"\"\"%\"xG!\"% *$F*F(F(F)-%\"yG6#F*F)F)*&,,!\"\"F)*$F*\"\"&\"\")*$F*\"\"%\"#7*$F*\"\" $F7F,F7F)-%%diffG6$F-F*F)F)*&,.*$F*\"\"'F7F3F7F6F)F9F+F,F+F*F2F)-F<6$F ;F*F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "gfun[`diffeq+dif feq`](deq1,deq2,y(x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(*&,(\"\"# \"\"\"%\"xG!\"%*$F(F&F&F'-%\"yG6#F(F'F'*&,,!\"\"F'*$F(\"\"&\"\")*$F(\" \"%\"#7*$F(\"\"$F5F*F5F'-%%diffG6$F+F(F'F'*&,.*$F(\"\"'F5F1F5F4F'F7F)F *F)F(F0F'-F:6$F9F(F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "g fun[diffeqtorec](\",y(x),u(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<(, .*&,&*$%\"nG\"\"#\"\"%F(F*\"\"\"-%\"uG6#F(F+F+*&,(F(\"#;\"#7F+F'F*F+-F -6#,&F(F+F+F+F+F+*&,(F'F+F(\"\"(F2F+F+-F-6#,&F(F+F)F+F+F+*&,(!#;F+F(F> F'!\"%F+-F-6#,&F(F+\"\"$F+F+F+*&,(!#YF+F'F?F(!#GF+-F-6#,&F(F+F*F+F+F+* &,(F(!#5!#DF+F'!\"\"F+-F-6#,&F(F+\"\"&F+F+F+/-F-6#F)\"\"!/-F-6#F*&%#_C G6#FW/-F-6#FC,$Fen#F+F)/-F-Fgn,$Fen#\"\"*F*/-F-6#F+,$Fen#FaoF)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "and thus the first five zeroes of the Taylor expansion of the left-hand side of the functional equation conclude the proof." }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Conclu sion" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "These results are very goo d examples of the use of " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 384 ": experiments first lead to conjecture a general form for the sol ution to a problem and then a completely different process leads to a \+ proof. However, the apparent ease with which the problems treated here are solved using gfun hides the preliminary work which led to the for m under which this approach could work. For example this approach does not seem to work for higher values of " }{XPPEDIT 18 0 "N" "I\"NG6\" " }{TEXT -1 36 ", where similar results might exist." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Bibliography" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "[1] Arithmetic-Goemetric Means Revisited. Jonathan M. Bor wein, Petr Lisonek and John A. Macdonald. " }{TEXT 260 9 "MapleTech" } {TEXT -1 2 ", " }{TEXT 261 3 "4-1" }{TEXT -1 19 ", pp. 20-27 (1997)." }}}}}{MARK "0 4 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 } f." }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Conclu sion" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "These results are very goo d examples of the use of " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 384 ": experiments first lead to conjecture a general form for the sol ution to a problem and then a completely different process leads to a \+ proof. However, the apparent ease with which the problems treated here are solved using gfun hides the preliminary work which led to the for m under which this approach could work. For example this approach does not seem to work for higher values of " }{XPPEDIT 18 0 "N" "I\"NG6\" " }{TEXT -1 36 ", where similar results might exist." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Bibliography" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "[1] Arithmetic-Goemetric Means Revisited. Jonathan M. Bor wein, Petview;+cf!kyEviralvirtual !>9Dviruvisual visualisEvoid /i ԡvol#'Ayjvolk}volum!kwa bwabwabawabbwait ?waitg?waitgf? waitmarkovg?walkwant# ?Owarn2(?7pNSwastwayS6+ !.!>%f!k7pyxEways/!kEwdegcwes':+#] bf!kAy}Wx3E7platexmathstylsetreadoptionmadmathformatcallsequencoptvaludescriptfunctallowconversmaplexpressfirstreturnlistknownsecondrespthirdforminvokatwithwarnprotectnameredefinunprotectwidthposintmaximallinelargspannoverseveralbackslashquadlocalusfrwaycommadisplayindentnonnegintnumbmarkatbeginnfloatstylfloateithinlinscientificnotationintegerrepresentatusedlogarithmabsolutliesbetweenelementnotatotherwisnotrailingzerobooleantrailzeromayremovtruealsoplargconstructperformdirectapplevennohadgivenanyalgebraicpolynomialsuchthusselectalgebraicsubnumersimilarproductdiffeqapplihavecoefficizeroconcludcubicexpressalsoexistagainleadtermlucksituatfindbeforroutinfiveconclusthesverygoodexamplexperimentconjecturgeneralproblemcompletedifferproceshowevappareasetreatherehidepreliminarworkledunderseemhighvalumightbibliographgoemetricrevisitjonathanpetrlisonekjohnmacdonaldmapletechpptranslatintoplaycentralletoodifficultanalyticneighborhoodoriginuniqusolutquadratictheoremequivalstatnowusefirststepcomputseriexpanspossiblclosformturnoutdoessatisffuneqopmethodundeterminatcoefficientsoldosolvunappodguessdifferentialsatisfideqseriestodiffeqogfdsolvunabldeducrecurrenctaylorsolutiondiffeqtorecorderlinearfoundrsolvhencsuminfinitproofconsistobviousourapproachusingclosurequationimplementlefthandsidereduconlycompatiblinitialcondituψ{VERSION 2 3 "DEC ALPHA UNIX" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "2 D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 " " 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{P STYLE "Norma l" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } 0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 ""  0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 256 18 "PATTERNS IN WORDS\n" }}{PARA 257 "" 0 "" {TEXT 260 11 "Bruno Salvy" }}{PARA 258 "" 0 "" {TEXT -1 29 "(Version of February 7, 1997)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 23 "This worksheet applies " }{HYPERLNK 17 "c ombstruct" 2 "combstruct" "" }{TEXT -1 5 " and " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 766 " to a simple combinatorial model of a probl em from computational biology and the study of DNA sequences. The DNA \+ can be viewed as a long text on an alphabet of four letters (A,C,G,T) . Large fragments of this text are tabulated. In particular, there are huge bases of genes, a gene being a few thousand letters long. Given \+ a short word, it is interesting to determine whether its number of occ urrences in a gene (or a virus) is far away from the most probable num ber of occurrences. If this number of occurrences is very improbable, \+ then this particular word m ay have a biological function. \n\nThe comb inatorial model is rational. The text is a word on the alphabet (a,b,c ,d) where all words of the same length are equiprobable. The probabili ty that a pattern occurs " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 72 " times in the text depends on the way the pattern overlaps with itsel f. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "libname:=`/net/blagny /algo/maple/5.4`,libname:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(combstruct): with(gfun):" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 264 70 "Specification and univariate generating functions for the patt ern abab" }}{PARA 0 "" 0 "" {TEXT -1 190 "Working over the alphabet (a ,b,c,d), we first concentrate on a specific pattern (abab). To attack \+ problems related to occurrences of this pattern in words using combstr uct, we first write a " }{TEXT 261 7 "grammar" }{TEXT -1 293 " which d escribes a corresponding automaton.This grammar recognizes all the wor ds on (a,b,c,d). It is written in such a way that a mar k (named Mark) \+ is present in a word everytime the pattern abab occurs. Then counting \+ the number of marks in a word gives the number of occurences of abab i n it." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 371 "G:=\{w=Union(Epsil on,Prod(a,wa),Prod(b,w),\n Prod(c,w),Prod(d,w)),\n wa=Un ion(Epsilon,Prod(a,wa),Prod(b,wab),\n Prod(c,w),Prod(d,w)), \n wab=Union(Epsilon,Prod(a,waba),Prod(b,w),\n Prod(c,w) ,Prod(d,w)),\n waba=Union(Epsilon,Prod(a,wa),Prod(b,Prod(Mark,w)), \n Prod(c,w),Prod(d,w)),\n Mark=Epsilon,a=Atom,b=Atom,c= Atom,d=Atom\}:" }}}{PARA 0 "" 0 "" {TEXT -1 20 "We use the function " }{HYPERLNK 17 "combstruct[count]" 2 "combstruct[count]" "" }{TEXT -1 45 " to check that the number of words of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "4^n" ")\"\"%%\"nG" } {TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "count([w,G, unlabelled],size=10),4^10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"(w&[5F #" }}}{EXCH G {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "count([w,G,unlabelled] ,size=20),4^20;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\".wxi6&*4\"F#" }}} {PARA 0 "" 0 "" {TEXT -1 23 "It is also possible to " }{TEXT 257 5 "pr ove" }{TEXT -1 64 " this by computing the generating function of the l anguage with " }{HYPERLNK 17 "combstruct[gfsolve]" 2 "combstruct[gfsol ve]" "" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "gf solve(G,unlabelled,z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<+/-%$wabG6# %\"zG,$*$,&F(\"\"%!\"\"\"\"\"F-F-/-%#waGF'F)/-%\"wGF'F)/-%%wabaGF'F)/- %\"cGF'F(/-%\"dGF'F(/-%\"aGF'F(/-%\"bGF'F(/-%%MarkGF'F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(\",w(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$,&%\"zG\"\"%!\"\"\"\"\"F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 19 "The coefficient of " }{XPPEDIT 18 0 "z^n" ")%\"zG%\"nG" } {TEXT -1 67 " in the Taylor expansion of this generating function is t he number " }{XPPEDIT 18 0 "4^n" ")\"\"%%\"nG" }{TEXT -1 20 " of words of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 127 " in the lang uage, which confirms the correctness of our grammar.\n\nHere are a few typical words of the language obtained by the " }{TEXT 262 24 "unifor m random generator" }{TEXT -1 13 " provided by " }{HYPERLNK 17 "combst ruct[draw]" 2 "combstruct[draw]" "" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 76 "to 20 do eval(subs(Prod=proc() args end,draw ([w,G,unlabelled],size=30))) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A% \"cG%\"dG%\"aGF$F$F%%\"bGF#F&F$F#F&F%F$F&F%F#F&F#F$F#F%F%F#F%F%F$F&F%F #%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"bGF#%\"cGF$%\"dG%\"a GF#F#F#F#F$F&F#F$F&F#F%F$F&F&F$F%F$F%F&F$F$F&F&F&%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"dGF#%\"cG%\"aG%\"bGF&F%F$F$F$F&F&F$F%F#F& F$F&F%F&F#F&F#F$F#F%F&F%F$F%%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"bG%\"aG%\"cGF$%\"dGF%F$F$F$F&F#F#F#F%F%F&F$F&F#F%F#F$F#F#F$F% F#F%F&F#%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"dG%\"aG%\"cGF #F%F#%\"bGF#F&F$F%F#F#F&F#F&F%F#F$F%F#F#F%F$F#F$F#F$F#F#%(EpsilonG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6A%\"cG%\"bGF$F$F$F#%\"aGF#F%F$%\"dGF$F$ F$F%F$F$F%F$F#F%F#F$F#F%F%F$F$F$F%%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"bG%\"cGF#%\"dGF%%\"aGF$F$F&F$F$F&F%F#F$F$F&F%F$F#F%F &F&F#F$F&F&F&F&F%%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"cG% \"aG%\"bGF#%\"dGF#F%F#F$F$F%F%F$F#F$F&F#F&F&F&F%F&F&F&F#F&F%F$F&F%%(Ep silonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"cG%\"dGF$F$%\"bGF$F%%\"aG F$F#F&F$F$F%F%F&F#F$F#F&F%F%F#F&F$F&F&F%F%F%%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"aGF#%\"bGF$%\"dGF#F#F$F#F%F%F%F%F#F#F#F#%\"cGF% F$F&F$F$F$F#F#F&F#F#F%%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A% \"aG%\"bG%\"cG%\"dGF#F$F$F%F&F%F&F%F#F%F%F%F&F%F&F#F&F#F#F&F&F#F$F#F%F #%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"aG%\"dG%\"bGF%F$F#F$ F$F#F%F$F#F#F$%\"cGF#F$F&F$F$F$F%F$F&F#F$F&F%F%F$%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6B%\"dG%\"bGF$%\"aGF$F%F#F$F%F$F%F$%%MarkGF#F$ F%%\"cGF$F#F'F#F#F$F#F#F'F%F'F'F$F#%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"bG%\"dGF#%\"aGF%F#F$F$%\"cGF#F&F#F&F#F%F%F&F#F#F$F$F $F&F#F%F%F#F$F#F#%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"bGF# F#%\"cG%\"aGF$F$F%F#%\"dGF&F$F#F%F%F#F#F#F$F&F%F%F%F%F&F&F$F#F$F$%(Eps ilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"aG%\"bGF#%\"cGF$F%%\"dGF$F $F$F#F%F&F$F$F&F$F$F&F%F#F#F$F%F$F%F%F$F&F%%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"aG%\"cG%\"dG%\"bGF$F#F#F&F&F&F&F&F&F&F#F&F$F&F& F#F#F#F$F%F#F&F%F%F&F%%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6B% \"bG%\"dGF#%\"aGF%F#F$F#F#%\"cGF&F#F&F&F&F&F&F$F&F&F$F%F&F%F#F%F#%%Mar kGF&F%F#%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"dG%\"bGF$%\"a G%\"cGF&F#F#F%F$F&F&F&F&F&F&F%F$F&F&F%F$F$F%F&F&F&F$F$F&%(EpsilonG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6A%\"bG%\"cGF$F$F$F$F#%\"dGF$F%F#F#F$F%F %F%F$%\"aGF#F#F$F&F&F&F&F$F&F#F%F&%(EpsilonG" }}}{PARA 0 "" 0 "" {TEXT -1 109 "Some of these words countain the pattern abab, as indica ted by the letter `Mark' right after its occurrence. " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 265 30 "Bivariate generating functions" }}{PARA 0 "" 0 "" {TEXT -1 50 "From the grammar specification above, the comma nd " }{HYPERLNK 17 "combstruct[gfsolve]" 2 "combstruct[gfsolve]" "" } {TEXT -1 29 " can also be used to derive " }{TEXT 263 12 "multivariat e" }{TEXT -1 96 " generating functions. From this, it is easy to compu te the probability that the pattern occurs " }{XPPEDIT 18 0 "k" "I\"kG 6\"" }{TEXT -1 34 " times in a random word of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 109 ", the expectation of the number of occur rences of the pattern in such a word, and the corresponding variance. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "gfsolve(G,unlabelled,z,[ [u,Mark]]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<+/-%\"cG6$%\"zG%\"uGF( /-%\"dGF'F(/-%\"aGF'F(/-%\"bGF'F(/-%%MarkGF'F)/-%%wabaGF',$*&,.\"\"\"F F=F@FE*$F(FEF=*&F(FEF)FFF " 0 "" {MPLTEXT 1 0 20 "sol:=subs(\",w(z,u)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG,$*&,&\"\"\"F(*$%\"zG\"\"#F (F(,.!\"\"F(F*\"\"%F)F-*$F*\"\"$F.*$F*F.F-*&F*F.%\"uGF(F(F-F-" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The coefficient of " }{XPPEDIT 18 0 "z^n*u^k" "*&)%\"zG%\"nG\"\"\")%\"uG%\"kGF&" }{TEXT -1 44 " in the T aylor series of this expression at " }{XPPEDIT 18 0 "z=0" "/%\"zG\"\"! " }{TEXT -1 34 " is the number of words of length " }{XPPEDIT 18 0 "n " "I\"nG6\"" }{TEXT -1 31 " where the pattern abab occurs " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 18 " times. Replacing " }{XPPEDIT 18 0 " z" "I\"zG6\"" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "z/4" "*&%\"zG\"\"\"\" \"%!\"\"" }{TEXT -1 21 " directly yields the " }{TEXT 258 31 "probabil ity generating function" }{TEXT -1 30 " under the uniform model (see \+ " }{HYPERLNK 17 "below" 1 "" "biased" }{TEXT -1 21 " for a biased mode l):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "GF:=normal(subs(z=z/4,sol)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GFG,$*&,&\"#;\"\"\"*$%\"zG\"\"# F)F),.!$c#F)F+\"$c#F*!#;*$F+\"\"$F(*$F+\"\"%!\"\"*&F+F4%\"uGF)F)F5F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Here are the first coefficients :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "S:=map(normal,series(GF,u));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"SG+1%\"uG,$*&,&\"#;\"\"\"*$%\"zG \"\"#F+F+,,\"$c#F+F-!$c#F,F**$F-\"\"$!#;*$F-\"\"%F+!\"\"F*\"\"!,$*(F)F +F-F6F/!\"#F*\"\"\",$*(F)F+F-\"\")F/!\"$F*\"\"#,$*(F)F+F-\"#7F/!\"%F* \"\"$,$*(F)F+F-F*F/!\"&F*\"\"%,$*(F)F+F-\"#?F/!\"'F*\"\"&-%\"OG6#F+\" \"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "For instance, the coeffici ent of " }{XPPEDIT 18 0 "u^0" "*$%\"uG\"\"!" }{TEXT -1 77 " in this se ries gives the probabilities that the pattern abab does not occur:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "series(coeff(S,u,0),z,31); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+]o%\"zG\"\"\"\"\"!F%\"\"\"F%\"\"# F%\"\"$#\"$b#\"$c#\"\"%#\"$F\"\"$G\"\"\"&#\"%\\S\"%'4%\"\"'#\"%SF4\"\")#\"&lS'\"&Ob'\"\"*#\"(J7-\"\"(w&[5\"#5#\"'fVD\"'W@E \"#6#\")\"[=i\"\");sx;\"#7#\")6#eh\"FK\"#8#\")l\")4;FK\"#9#\"*PnIG\"\" *Gx@M\"\"#:#\"+Ryb!4%\"+'Hn\\H%\"#;#\"+(Ryw.#\"+[O[Z@\"#<#\",4eRj\\'\" ,OnZ>(o\"#=#\",Xe\\!=;\",%=p)zr\"\"#>#\",r!o9[kF[o\"#?#\".rZapy-\"\".w xi6&*4\"\"#@#\"/rac(z%Q;\"/;W/'=#f<\"#A#\".23p)[S?\"._bDB!*>#\"#B#\"0D ]3k>@g#\"0c1rw\\Z\"G\"#C#\"0p#zD)\\Cf#Fbp\"#D#\"0pD]%f\"Ge#Fbp\"#E#\"1 ,^+YrGH5\"1CE%o!***e7\"\"#F#\"2R'o%*ez&Hc'\"2Oz#z.%fd?(\"#G#\"2N]Xnd%G pK\"2oR'*=qzGg$\"#H#\"4TyP,oN$GU5\"4wp%og/:#H:\"\"#I-%\"OG6#F%\"#J" }} }{PARA 0 "" 0 "" {TEXT -1 144 "Thus the random draws of words of lengt h 30 that we did before are typical: the probability that the pattern \+ does not occur in such a word being" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "coeff(\",z,30)=evalf(coeff(\",z,30));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"4TyP,oN$GU5\"4wp%og/:#H:\"$\"+QqOS!*!#5" }}} {PARA 0 "" 0 "" {TEXT -1 135 "The expected number of occurrences is ob tained very directly from the bivariate generating function GF. Here i s its generating function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "mom1:=factor(subs(u=1,diff(GF,u)));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%%mom1G,$*(%\"zG\"\"%,&F'\"\"\"!\"\"F*!\"#,&\"#;F**$F'\"\"#F*F+#F*F ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "smom1:=series(\",z,31) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&smom1G+en%\"zG#\"\"\"\"$c#\"\" %#F(\"$G\"\"\"&#\"#Z\"%'4%\"\"'#\"#J\"%[?\"\"(#\"%L7\"&Ob'\"\")#\"$P( \"&oF$\"\"*#\"&Ru#\"(w&[5\"#5#\"&Zc\"\"')GC&\"#6#\"'&Qi&\");sx;\"#7#\" 'L?J\"(3')Q)\"#8#\")^>(4\"\"*caVo#\"#9#\"(B%zf\"*Gx@M\"\"#:#\"*d=82#\" +'Hn\\H%\"#;#\"**3h96\"+[O[Z@\"#<#\"+$)**R>Q\",OnZ>(o\"#=#\"+fD-O?\",o $Q(fV$\"#>#\",\\S/&>p\".wxi6&*4\"\"#?#\",0Jo=m$\"-))Q\"ev\\&\"#@#\".NR ,vkB\"\"/;W/'=#f<\"#A#\"-bU3w0l\".3A-$4'z)\"#B#\"/h$p\"pK&=#\"0c1rw\\Z \"G\"#C#\"/\"G@=0W9\"\"0G`N)[P29\"#D#\"02A&)4qw#Q\"1'\\qti*f.X\"#E#\"0 6\"1[_o8)*z^A\"#F#\"1$>9zS2Tl'\"2Oz#z.%fd?(\"#G#\"1*3JD3V\\6\"4wp%og/:#H:\"\"#I-%\"OG6#F(\"#J" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(\");" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#+en%\"zG$\"+++D1R!#7\"\"%$\"+++]7yF'\"\"&$\"+Q4YZ6!#6 \"\"'$\"+v=n8:F.\"\"($\"+\"p39)=F.\"\")$\"+3b9\\AF.\"\"*$\"+dpy;EF.\"# 5$\"+1%GW)HF.\"#6$\"+:e2_LF.\"#7$\"+DKs>PF.\"#8$\"+i-P(3%F.\"#9$\"+)H< ]X%F.\"#:$\"+fVmA[F.\"#;$\"+>9J!>&F.\"#<$\"+y%ezb&F.\"#=$\"+ObgDfF.\"# >$\"+&f_KH'F.\"#?$\"+a'**3m'F.\"#@$\"+8naGqF.\"#A$\"+sP>'R(F.\"#B$\"+I 3%Qw(F.\"#C$\"+*)y[J\")F.\"#D$\"+[\\8*\\)F.\"#E$\"+2?ym))F.\"#F$\"+m!H WB*F.\"#G$\"+Dh2-'*F.\"#H$\"+$=B(p**F.\"#I-%\"OG6#\"\"\"\"#J" }}} {PARA 0 "" 0 "" {TEXT -1 106 "Thus in a sequence of 20 draws as above, we can expect the following number of occurrences of the pattern:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "20*coeff(\",z,30);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+PY%R*>!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "The variances are computed as easily as the expectations: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "mom2:=factor(subs(u=1,d iff(u*diff(GF,u),u)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%mom2G,$** %\"zG\"\"%,,\"$G\"\"\"\"F'!$G\"*$F'\"\"#\"\")*$F'\"\"$!\")*$F'F(F+F+,& F'F+!\"\"F+!\"$,&\"#;F+F-F+!\"##F5F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "evalf(series(mom2-add(coeff(smom1,z,i)^2*z^i,i=0..30) ,z,31));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+en%\"zG$\"+67*4*Q!#7\"\"% $\"+W[Y^xF'\"\"&$\"+rUHM6!#6\"\"'$\"+])f2\\\"F.\"\"($\"+jM1\\=F.\"\")$ \"+EUr2AF.\"\"*$\"+%HSic#F.\"#5$\"+YMtCHF.\"#6$\"+/]B$G$F.\"#7$\"+@$R< k$F.\"#8$\"+uIC+SF.\"#9$\"+6mueVF.\"#:$\"+'=]sr%F.\"#;$\"+xPvv]F.\"#<$ \"+ltDMaF.\"#=$\"+_4w#z&F.\"#>$\"+SXE^hF.\"#?$\"+F\"o(4lF.\"#@$\"+:!*F.\"#G$\"+Fozx$*F.\"#H$\"+9/IO(*F.\"#I-%\"O G6#\"\"\"\"#J" }}}{PARA 0 "" 0 "" {TEXT -1 19 "The coefficient of " } {XPPEDIT 18 0 "z^n" ")%\"zG%\"nG" }{TEXT -1 96 " in this series is the variance of the number of occurrences of the pattern in a word of len gth " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 1 "." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 266 27 "Fixed number of occurrences" }}{PARA 0 "" 0 " " {TEXT -1 51 "We now consider the probabilities that abab occurs " } {XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 12 " times, for " }{XPPEDIT 18 0 "k=0..5" "/%\"kG;\"\"!\"\"&" }{TEXT -1 8 ". Using " }{HYPERLNK 17 "g fun" 2 "gfun" "" }{TEXT -1 64 ", we can compute these probabilities fo r random words of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 7 " , with " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 23 " up to a few thou sands." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "maxnb:=5:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "for i from 0 to maxnb do pro ba[i]:=coeff(S,u,i) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&probaG6 #\"\"!,$*&,&\"#;\"\"\"*$%\"zG\"\"#F,F,,,\"$c#F,F.!$c#F-F+*$F.\"\"$!#;* $F.\"\"%F,!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&probaG6#\"\" \",$*(,&\"#;F'*$%\"zG\"\"#F'F'F-\"\"%,,\"$c#F'F-!$c#F,F+*$F-\"\"$!#;*$ F-F/F'!\"#F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&probaG6#\"\"#,$*(, &\"#;\"\"\"*$%\"zGF'F,F,F.\"\"),,\"$c#F,F.!$c#F-F+*$F.\"\"$!#;*$F.\"\" %F,!\"$F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&probaG6#\"\"$,$*(,&\" #;\"\"\"*$%\"zG\"\"#F,F,F.\"#7,,\"$c#F,F.!$c#F-F+*$F.F'!#;*$F.\"\"%F,! \"%F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&probaG6#\"\"%,$*(,&\"#;\" \"\"*$%\"zG\"\"#F,F,F.F+,,\"$c#F,F.!$c#F-F+*$F.\"\"$!#;*$F.F'F,!\"&F+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&probaG6#\"\"&,$*(,&\"#;\"\"\"* $%\"zG\"\"#F,F,F.\"#?,,\"$c#F,F.!$c#F-F+*$F.\"\"$!#;*$F.\"\"%F,!\"'F+ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 231 "Since we want to investigate these probabilities for texts of large size (a typical gene is a few \+ thousand letters long), we need the Taylor expansions of these rationa l functions for very large orders. These can be computed using " } {HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 31 ", which will first compu te the " }{TEXT 259 6 "linear" }{TEXT -1 52 " recurrence satisfied by \+ these Taylor coefficients (" }{HYPERLNK 17 "gfun[diffeqtorec]" 2 "gfun [diffeqtorec]" "" }{TEXT -1 73 "), and then exploit these recurrences \+ to compute the series efficiently (" }{HYPERLNK 17 "gfun[rectoproc]"  2 "gfun[rectoproc]" "" }{TEXT -1 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "for i from 0 to maxnb do \n rec:=diffeqtorec(y(z )-proba[i],y(z),u(n));\n print(i,rec);\n rec:=select(has,rec,n) un ion \{seq(op(1,i)=evalf(op(2,i)),i=remove(has,rec,n))\};\n P[i]:=rec toproc(rec,u(n),remember) \nod:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\" \"!<',,-%\"uG6#%\"nG\"\"\"-F'6#,&F)F*F*F*!#;-F'6#,&F)F*\"\"#F*\"#;-F'6 #,&F)F*\"\"$F*!$c#-F'6#,&F)F*\"\"%F*\"$c#/-F'6#F2F*/-F'6#F7F*/-F'6#F#F */-F'6#F*F*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$\"\"\"<+/-%\"uG6#F#\"\" !/-F'6#\"\"$F)/-F'6#\"\"#F),4-F'6#%\"nGF#-F'6#,&F5F#F#F#!#K-F'6#,&F5F# F1F#\"$)G-F'6#,&F5F#F-F#!%C5-F'6#,&F5F#\"\"%F#\"%g*)-F'6#,&F5F#\"\"&F# !&%Q;-F'6#,&F5F#\"\"'F#\"&GP(-F'6#,&F5F#\"\"(F#!'s58-F'6#,&F5F#\"\")F# \"&Ob'/-F'6#F)F)/-F'6#FE#F#\"$c#/-F'6#FJ#F#\"$G\"/-F'6#FO#\"#Z\"%'4%/- F'6#FT#\"#J\"%[?" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$\"\"#\"-F'6#,&FHF)\"\"'F)\"(obS(-F'6#,&FHF)\"\"(F)!)+!oF$-F' 6#,&FHF)FDF)\"*SI#[;-F'6#,&FHF)F8F)!*+!)GC&-F'6#,&FHF)F04\"-F'6#,&FHF)\"#8F)!,![O [Z@-F'6#,&FHF)\"#9F)\",+caVo#-F'6#,&FHF)\"#:F)!,%=p)zr\"-F'6#,&FHF)\"# ;F)\"+'Hn\\H%/-F'6#FXF*/-F'6#FgnF*/-F'6#F\\oF*/-F'6#FaoF*/-F'6#F`q#\"$ d\"\"*caVo#/-F'6#Feq#\"#x\")k)3r'/-F'6#Ffp#F)\");sx;/-F'6#F[q#F)\"(/V> %" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$\"\"%<7/-%\"uG6#\"\"\"\"\"!/-F'6# \"\"$F*/-F'6#\"\"#F*/-F'6#F*F*/-F'6#\"\"*F*/-F'6#\"#5F*/-F'6#\"#6F*/-F '6#\"\")F*/-F'6#\"#8F*/-F'6#\"#9F*/-F'6#\"#:F*/-F'6#\"#7F*,L-F'6#%\"nG F)-F'6#,&FYF)F)F)!#!)-F'6#,&FYF)F2F)\"%SE-F'6#,&FYF)F.F)!>%-F'6#,&FY F)F#F)\"'?j`-F'6#,&FYF)\"\"&F)!('47Y-F'6#,&FYF)\"\"'F)\")gp&[$-F'6#,&F YF)\"\"(F)!*!)[B=#-F'6#,&FYF)FEF)\"+?FQc6-F'6#,&FYF)F9F)!+SUv()e-F'6#, &FYF)F=F)\",ca.1K#-F'6#,&FYF)FAF)!,Sy1/U*-F'6#,&FYF)FUF)\"-?j(R.'H-F'6 #,&FYF)FIF)!-![o+*Q*)-F'6#,&FYF)FMF)\".g0t&Q%G#-F'6#,&FYF)FQF)!.'HvJ8O [-F'6#,&FYF)\"#;F)\".?^[cz**)-F'6#,&FYF)\"#.Jr7-F'6#,&FYF)\"#= F)\"/S9m8(Q8\"-F'6#,&FYF)\"#>F)!.!))Q\"ev\\&-F'6#,&FYF)\"#?F)\".wxi6&* 4\"/-F'6#F#F*/-F'6#FgoF*/-F'6#F\\pF*/-F'6#FapF*/-F'6#Ffr#F)\"+'Hn\\H%/ -F'6#F[s#FgoF\\u/-F'6#F`s#\"#f\",%=p)zr\"/-F'6#Fes#\"$N\"Ffu" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$\"\"&<;/-%\"uG6#\"\"\"\"\"!/-F'6#\"\"$F*/-F' 6#\"\"#F*/-F'6#F*F*/-F'6#\"\"*F*/-F'6#\"#5F*/-F'6#\"#6F*/-F'6#\"\")F*/ -F'6#\"#8F*/-F'6#\"#9F*/-F'6#\"#:F*/-F'6#\"#7F*,T-F'6#,&%\"nGF)FIF)!/; s$*3de9-F'6#,&FZF)FMF)\"/O6*o:!Q^-F'6#,&FZF)FQF)!0'*[E&[*eb\"-F'6#,&FZ F)\"#;F)\"0;wW9oaF%-F'6#,&FZF)F)F)!#'*-F'6#,&FZF)F2F)\"%OR-F'6#,&FZF)F .F)!&O6*-F'6#,&FZF)\"\"%F)\"(cqN\"-F'6#,&FZF)\"#=F)\"1c5Y3&o:=#-F'6#,& FZF)\"#>F)!1wp#4e`=$R-F'6#,&FZF)\"#?F)\"1w0%)Q6^Ge-F'6#,&FZF)\"# " 0 "" {MPLTEXT 1 0 52 "Digits:=30:for i from 0 to maxnb do i,P[i](1000) od; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!$\"?H)f4f^R0U>$R8EUC!#J" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"$\"?gier+tR,%\\+Vs;=*!#J" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"#$\"?C=5L0q8+%e<-kTr\"!#I" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"$$\"?k@mY![0hd;Y0$))=@!#I" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"%$\"?]_Mx7(R*[%*oXK%3&>!#I" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"&$\"?CEykKv=B\">0`epU\"!#I" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "The following picture then shows h ow these probabilities evolve with the length of the word:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "plots[display](\{seq(plot([seq([10* i,P[j](10*i)],i=1..100)]),j=0..maxnb)\});" }}{PARA 13 "" 1 "" {INLPLOT "6(-%'CURVESG6$7`q7$$\"#5\"\"!$\"+u&QX'*)!#97$$\"#?F*$\"?%Re) zu!of-B1RtNC\"!#K7$$\"#IF*$\"?l3;/S#)3\">=fBD$)f$F37$$\"#SF*$\">/N&eJB H3Fn\"o#e7$$\"#gF*$\"?%\\?0.=^m:9' 4ZUb;F>7$$\"#qF*$\"?$Hd;mTp^[)[=HqWAF>7$$\"#!)F*$\"?6&fqK$fMl=$GJ$R&*G F>7$$\"#!*F*$\"?;1vjj@b/\\#o%3K)f$F>7$$\"$+\"F*$\"?vF(*f(eO*f%pldT]M%F >7$$\"$5\"F*$\">`.>e3t;F\"zoizF^!#I7$$\"$?\"F*$\">9P$yt:6P^B9mYRfF\\o7 $$\"$I\"F*$\">*pSG&RDPWv.\"R`tnF\\o7$$\"$S\"F*$\">`v>$3M1;_t%3SSi(F\\o 7$$\"$]\"F*$\">'[%Qm_5?9hk$4b&[)F\\o7$$\"$g\"F*$\">#Q.,0T9c[&)z97`$*F \\o7$$\"$q\"F*$\"?h3m!eP8]fKzKEA-\"F\\o7$$\"$!=F*$\"?$o+hCym0IsqO\"*)3 6F\\o7$$\"$!>F*$\"?L@\"G58*=1G#pnS\\>\"F\\o7$$\"$+#F*$\"?>qw['*zXUz*oh W+G\"F\\o7$$\"$5#F*$\"?.Wc)*4c89HVCz!RO\"F\\o7$$\"$?#F*$\"?yK$*4gk'Ge: nHniW\"F\\o7$$\"$I#F*$\"?)Hf!4!H'oO^*e6*)o_\"F\\o7$$\"$S#F*$\"?D1%p:$Q *G8\\+Jnbg\"F\\o7$$\"$]#F*$\"?I'*Q1I<3Z$zDM@@o\"F\\o7$$\"$g#F*$\"?/\\8 .*QMZ`xIK%RcO,?@7I()\\0L7DG=F\\o7$$\"$!GF*$\"?BLWD 9e*oZl-Exv*=F\\o7$$\"$!HF*$\"?$\\/Z_u'pH1nNiFk>F\\o7$$\"$+$F*$\"?!G@)3 K7X32Sn\"p#G?F\\o7$$\"$5$F*$\"?eSHxo=JXHNmG\\*3#F\\o7$$\"$?$F*$\"?A(yy l(**fSYMe')*y9#F\\o7$$\"$I$F*$\"?db_**))zT#*)**z<^M?#F\\o7$$\"$S$F*$\" ?<'HPO#*ovn(zRs7cAF\\o7$$\"$]$F*$\"?a%ohoj\"Qoc%>y9fI#F\ \o7$$\"$g$F*$ \"?%*pVGj)Hw@p*y=\"GN#F\\o7$$\"$q$F*$\"?*RKOS(=HB*=\">e#oR#F\\o7$$\"$! 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Th)e@)[hTLFK#=iZ)F-7$F^jl$\"?wI(49Vm*Hanp')4D%)F-7$Fcjl$\"?rk9)GTj0V?& HACt$)F-Fgjl" 2 236 216 216 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 17 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "The complexity of these compu tations grows only linearly with the number " }{XPPEDIT 18 0 "k" "I\"k G6\"" }{TEXT -1 95 " of occurrences under study. Other kinds of constr aints like number of occurrences larger than " }{XPPEDIT 18 0 "k" "I\" kG6\"" }{TEXT -1 11 " for fixed " }{XPPEDIT 18 0 "k" "I\"kG6\"" } {TEXT -1 12 " or between " }{XPPEDIT 18 0 "k[1]" "&%\"kG6#\"\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "k[2]" "&%\"kG6#\"\"#" }{TEXT -1 141 " also give rise to rational generating functions that can be extr acted from the generating function GF and thus can be treated the same way. " }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 267 14 "Other patterns" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 241 "All the above computation was der ived from the Egrammar describing the language, a mark being appended t o every occurrence of the pattern. It is actually easy to write a Mapl e procedure taking as input a word, and producing the corresponding " }{HYPERLNK 17 "combstruct grammar" 2 "combstruct[specification]" "" } {TEXT -1 94 ". Then the whole computation above can be reproduced for \+ any pattern completely automatically." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 561 "gengram:=proc(pattern::list(\{identical(a),identical (b),identical(c),identical(d)\}))local i, eq, letter, state, j;\nfor i to nops(pattern) do for letter in [a,b,c,d] do for j from 0 to i-1 do if [op(1..i-j,pattern)]=[op(j+1..i-1,pattern),letter] then state[lett er]:=cat(w,op(1..i-j,pattern)); break fi od; if j=i then state[letter] :=w fi od;eq[i]:=cat(w,op(1..i-1,pattern))=Union(Epsilon,seq(Prod(lett er,state[letter]),letter=[a,b,c,d])) od; subs(cat(w,op(pattern))=Prod( Mark,w),\{seq(eq[i],i=1..nops(pattern)),seq(letter=Atom,letter=[a,b,c, d]),Mark=Epsilon\}) end;" }}{PARA 12 "" 1 F"" {XPPMATH 20 "6#>%(gengram G:6#'%(patternG-%%listG6#<&-%*identicalG6#%\"aG-F.6#%\"bG-F.6#%\"cG-F. 6#%\"dG6'%\"iG%#eqG%'letterG%&stateG%\"jG6\"F@C$?(8$\"\"\"FD-%%nopsG6# 9$%%trueGC$?&8&7&F0F3F6F9FIC$?(8(\"\"!FD,&FCFD!\"\"FDFI@$/7#-%#opG6$;F D,&FCFDFPFSFH7$-FX6$;,&FPFDFDFDFRFHFLC$>&8'6#FL-%$catG6$%\"wGFW%&break G@$/FPFC>F]oFco>&8%6#FC/-Fao6$Fco-FX6$;FDFRFH-%&UnionG6$%(EpsilonG-%$s eqG6$-%%ProdG6$FLF]o/FLFM-%%subsG6$/-Fao6$Fco-FXFG-Fjp6$%%MarkGFco<%/F fqFep-Fgp6$/FL%%AtomGF\\q-Fgp6$Fio/FC;FDFEF@F@" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "Given a pattern, this procedure outputs the corresp onding combstruct grammar. Thus for instance, using the pattern abab, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "gengram([a,b,a,b]);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#<+/%\"wG-%&UnionG6'%(EpsilonG-%%ProdG6 $%\"aG%#waG-F+6$%\"bGF%-F+6$%\"cGF%-F+6$%\"dGF%/F.-F'6'F)F*-F+6$F1%$wa bGF2F5/F=-F'6'F)-F+6$F-%%wabaGF/F2F5/FC-F'6'F)F*-F+6$F1-F+6$%%MarkGF%F 2F5/FKF)/F-%%AtomG/F1FN/F4FN/F7FN" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 G 211 "we obtain the grammar we started with. It is now possible to stud y longer patterns easily. Here are the different states leading to the probability that the pattern abacab occurs twice in a word of length 5000:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "First the grammar is ge nerated" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "G:=gengram([a,b, a,c,a,b]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"GG<-/%\"wG-%&UnionG6 '%(EpsilonG-%%ProdG6$%\"aG%#waG-F-6$%\"bGF'-F-6$%\"cGF'-F-6$%\"dGF'/F0 -F)6'F+F,-F-6$F3%$wabGF4F7/F?-F)6'F+-F-6$F/%%wabaGF1F4F7/%%MarkGF+/F/% %AtomG/F3FI/F6FI/F9FI/FE-F)6'F+F,F=-F-6$F6%&wabacGF7/FR-F)6'F+-F-6$F/% 'wabacaGF1F4F7/FX-F)6'F+F,-F-6$F3-F-6$FGF'F4F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "then the bivariate generating functions are derived " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "gfsolve(G,unlabelled,z, [[u,Mark]]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<-/-%\"cG6$%\"zG%\"uGF (/-%\"dGF'F(/-%\"aGF'F(/-%\"bGF'F(/-%%MarkGF'F)/-%\"wGF',$*&,&\"\"\"F< *$F(\"\"%FFH=F@*$F(\"\"&F>*$F(\"\"'F@*&F(FDF)FF)F " 0 "" {MPLTEXT 1 0 29 "normal(subs(\",z=z/4,w(z,u)) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&\"$c#\"\"\"*$%\"zG\"\"%F'F ',.!%'4%F'F)\"%'4%F(!#;*$F)\"\"&\"#;*$F)\"\"'!\"\"*&F)F3%\"uGF'F'F4F. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "extracting a coefficient we \+ get the probability generating function of words with 2 occurrences of the pattern" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "coeff(serie s(\",u,3),u,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**,&\"$c#\"\"\"*$ %\"zG\"\"%F'F'F)\"#7,,\"%'4%F'F)!%'4%F(\"#;*$F)\"\"&!#I;*$F)\"\"'F'!\"# ,,F.F'F)F-F(F2F0F/F3!\"\"F7F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 " this gives rise to a linear recurrence satisfied by the Taylor coeffic ients" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "diffeqtorec(y(z)- \",y(z),u(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<5/-%\"uG6#\"\"\"\" \"!/-F&6#\"\"$F)/-F&6#\"\"#F)/-F&6#F)F)/-F&6#\"\"*F)/-F&6#\"#5F)/-F&6# \"#6F)/-F&6#\"\")F),H-F&6#%\"nGF(-F&6#,&FHF(F(F(!#[-F&6#,&FHF(F1F(\"$; )-F&6#,&FHF(F-F(!%Kc-F&6#,&FHF(\"\"%F(\"&cI\"-F&6#,&FHF(\"\"&F(!&wX#-F &6#,&FHF(\"\"'F(\"'+'4%-F&6#,&FHF(\"\"(F(!(g@$R-F&6#,&FHF(FDF(\"(+/$)* -F&6#,&FHF(F8F(!(%=P%*-F&6#,&FHF(FfT#-F&6#,&FHF(\"#9F(\"*oj I0)-F&6#,&FHF(\"#:F(!,OnZ>(o-F&6#,&FHF(\"#;F(\"-3-Veh?-F&6#,&FHF(\"#(o/-F&6#FXF)/-F&6#FgnF)/-F&6#F\\oF)/- F&6#FaoF)/-F&6#F_r#\"%&o#\"+[O[Z@/-F&6#Fjq#\"%>>Fgs/-F&6#Feq#Fgn\"(3') Q)/-F&6#F[q#F-\");sx;/-F&6#F`q#F-Fat/-F&6#Ffp#F(Fft" }}}{EXCHG {PARA 0 "" 0 "" {TJEXT -1 110 "As before, in order to speed up the computatio n, we change the initial conditions into floating point numbers:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=50:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "select(has,\"\",n) union \{seq(op(1 ,i)=evalf(op(2,i)),i=remove(has,\"\",n))\}:" }}}{EXCHG {PARA 0 "" 0 " " {HYPERLNK 17 "gfun[rectoproc]" 2 "gfun[rectoproc]" "" }{TEXT -1 64 " then produces a procedure to evaluate this sequence efficiently" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "p:=rectoproc(\",u(n),remembe r):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "for i from 0 by 500 \+ to 10000 do i,p(i) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!F#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"$+&$\"R#oj..Hv'*=N0T!zP<%p+!oVs6]J'! #^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+5$\"S$e%!#^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+?$\"R1/Lv5ZU> 29z5o8>)=/i&QSa#Rs!#]" }}{PARA 11 "" 1 "" {XPKPMATH 20 "6$\"%+D$\"S*G.l yI5$onRiZhRsv*)o)3XnuN+\"!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+I$ \"S$[&*fl-_#eE#fZSF$)*)3hE?yxv7G\"!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+N$\"S-$z#\\*)o87w/ty/y#\\8R$GFAHeX:!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+S$\"SWxl&*H2Vo>x$\\l1B\\@3AO/9U')y\"!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+X$\"S(=A15$4Z$ziuB/N^SARc9-B5a+#!#]" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+]$\"S.ZskEY])4(fA0&HkiCO$*>9#z,$># !#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+b$\"S'[tBBA\"QI'4KfLfiLlP-s V2`-N#!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+g$\"SNIud*Q`*>#GqT\"o Dh$3+>AG]VrZ#!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+l$\"S/Z%)yRJ-v 6?q(=)z`Etf;3fAiuD!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+q$\"S<\\n pmo)R/Sz*yG/-wrgdOs4EWE!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+v$\" S)[N*GAL60[Lu7&)\\>k\"f!)o#!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$\"%+!)$\"Sr%H,g.9m0ffd^^BkA)*\\z!R%z#3F!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+&)$\"SM-I%f!>(Hcv%ft#RHq+@pkw6?tq#!#]" }}{PARA 11 " " 1 "" {XPPMLATH 20 "6$\"%+!*$\"S:48c+o=d#)=j()e#y#)\\o%oU)G=wo#!#]" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+&*$\"SAa(Hqm+*Q`\"RQm8f%eyo2tTud^E !#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"&++\"$\"SCEWCN4uk-#zc*RJ$\\Uv KA)\\[^,E!#]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "The following pic ture shows the evolution of this probability with the length of the wo rd" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot([seq([100*i,p(10 0*i)],i=0..100)]);" }}{PARA 13 "" 1 "" {INLPLOT "6#-%'CURVESG6$7aq7$\" \"!F(7$$\"$+\"F($\"SYX.5hLk[*z&)eKd_4G4(z.;D)yK#!#`7$$\"$+#F($\"SsRs4l 03(>v%ziq$H*\\7[:rJBF47 $$\"$+%F($\"SIFF3G/kNRuu)pN_T_FLF\\HAq4%F47$$\"$+&F($\"R#oj..Hv'*=N0T! 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)o#Ffp7$$\"%+wF($\"S[&R?iD,Mo8gL'z)=zFx+.r#Ffp7$$\"%+$)F($\" S^K!QrE\"\\grj!>M?/9hRw)eZ$*35FFfp7$$\"%+%)F($\"S=1wv/z>mF.;%yx\"3^>oR wz#*34FFfp7$$\"%+&)F($\"SM-I%f!>(Hcv%ft#RHq+@pkw6?tq#Ffp7$$\"%+')F($\" SE:AxvjF^$)*zU>GGY%eR10r8![q#Ffp7$$\"%+()F($\"S>9.^:tL'3z8p%[9o&[mP14 \\_:q#Ffp7$$\"%+))F($\"SoQ6nHC$)[@6Rt?T.#fJH<0&Gf(p#Ffp7$$\"%+*)F($\"S Y%>N>%fl\"\\Z'=x\"4_$R7j " 0 "" {MPLTEXT 1 0 2 "G;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<-/% \"wG-%&UnionG6'%(EpsilonG-%%ProdG6$%\"aG%#waG-F+6$%\"bGF%-F+6$%\"cGF%- F+6$%\"dGF%/F.-F'6'F)F*-F+6$F1%$wabGRF2F5/F=-F'6'F)-F+6$F-%%wabaGF/F2F5 /%%MarkGF)/F-%%AtomG/F1FG/F4FG/F7FG/FC-F'6'F)F*F;-F+6$F4%&wabacGF5/FP- F'6'F)-F+6$F-%'wabacaGF/F2F5/FV-F'6'F)F*-F+6$F1-F+6$FEF%F2F5" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 327 "Marks are added by replacing a by Prod(Atom,Marka) everywhere it occurs, and similarly for the other le tters. There Marka, Markb, Markc and Markd have size 0 and do not modi fy the counting sequences and the related probabilities, but make it p ossible to extract multivariate generating functions which contain mor e information." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "Gprob:=s ubs(a=Prod(Atom,Marka),b=Prod(Atom,Markb),c=Prod(Atom,Markc),d=Prod(At om,Markd),G minus \{a=Atom,b=Atom,c=Atom,d=Atom\}) union \{Marka=Epsil on, Markb=Epsilon, Markc=Epsilon, Markd=Epsilon\};" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%&GprobG<-/%%MarkG%(EpsilonG/%&MarkaGF(/%&MarkbGF(/% &MarkcGF(/%&MarkdGF(/%'wabacaG-%&UnionG6'F(-%%ProdG6$-F76$%%AtomGF*%#w aG-F76$-F76$F;F,-F76$F'%\"wG-F76$-F76$F;F.FC-F76$-F76$F;F0FC/F<-F46'SF( F6-F76$F?%$wabGFDFH/%&wabacG-F46'F(-F76$F9F2-F76$F?FCFDFH/FQ-F46'F(-F7 6$F9%%wabaGFXFDFH/FC-F46'F(F6FXFDFH/Fin-F46'F(F6FO-F76$FFFSFH" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Here is the multivariate generatin g function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "gfsolve(Gpro b,unlabelled,z,[[u,Mark],[a,Marka],[b,Markb],[c,Markc],[d,Markd]]);" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#<-/-%#waG6(%\"zG%\"uG%\"aG%\"bG%\"cG% \"dG,$*&,*\"\"\"F1*,F(\"\"&F*\"\"#F+F4F,F1F)F1F1**F(\"\"%F,F1F+F1F*F4F 1**F(F3F,F1F+F4F*F4!\"\"F1,:*&F(F1F*F1F1*,F(\"\"'F,F1F*\"\"$F+F4F)F1F1 *,F(F3F,F1F-F1F+F1F*F4F1**F(F3F,F4F+F1F*F4F1F7F1F5F8**F(F3F,F1F*F=F+F1 F1**F(F " 0 "" {MPLTEXT 1 0 27 "GF:=su bs(\",w(z,u,a,b,c,d));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GFG,$*&,& \"\"\"F(**%\"zG\"\"%%\"cGF(%\"bGF(%\"aG\"\"#F(F(,:*&F*F(F.F(F(*,F*\"\" 'F,F(F.\"\"$F-F/%\"uGF(F(*,F*\"\"&F,F(%\"dGF(F-F(F.F/F(**F*F7F,F/F-F(F .F/F(**F*F7F,F(F-F/F.F/F(F)!\"\"**F*F7F,F(F.F4F-F(F(**F*F3F,F(F.F4F-F/ F;F;F(*&F*F(F-F(F(*&F*F(F,F(F(*&F*F(F8F(F(F;F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The coefficient of " }{XPPEDIT 18 0 "a^k*b^l*c^m*d^p *u^q*z^n" "*.)%\"aG%\"kG\"\"\")%\"bG%\"lGF&)%\"cG%\"mGF&)%\"dG%\"pGF&) %\"uG%\"qGF&)%\"zG%\"nGF&" }{TEXT -1 34 " in the Taylor expansion of G F in " }{XPPEDIT 18 0 "z" "I\"zG6\"" }{TEXT -1 34 " is the number of w ords of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 6 " with " } {XPPEDIT 18 0 "k " "I\"kG6\"" }{TEXT -1 30 " occurrences of the letter a, " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 37 " occurrences of the \+ letter b ... and " }{XPPEDIT 18 0 "q" "I\"qG6\"" }{TEXUT -1 93 " occurr ences of the pattern. The probability generating function is obtained \+ by substituting " }{XPPEDIT 18 0 "a,b,c,d" "6&%\"aG%\"bG%\"cG%\"dG" } {TEXT -1 148 " by the corresponding probability. Thus GF now takes int o account both the specific pattern and the biased probabilities of th e letters in the text." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "We can \+ again compute the probability that the pattern occurs twice in words o f length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 41 " and compare th is with the uniform model:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "normal(coeff(series(GF,u,3),u,2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*.,&\"\"\"F%**%\"zG\"\"%%\"cGF%%\"bGF%%\"aG\"\"#F%F%F'\"#7F)F,F+ \"\"'F*F(,8*&F'F%F+F%!\"\"F&F%*,F'\"\"&F)F%%\"dGF%F*F%F+F,F1**F'F3F)F, F*F%F+F,F1**F'F3F)F%F*F,F+F,F1*&F'F%F*F%F1**F'F3F)F%F+\"\"$F*F%F1**F'F .F)F%F+F9F*F,F%F%F%*&F'F%F)F%F1*&F'F%F4F%F1!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 "Again we compute a linear recurrence satisfied by t he TaylorV coefficients from which we produce an efficient procedure fo r their evaluation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "diff eqtorec(y(z)-\",y(z),u(n)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "nuprob:=rectoproc(subs([a=0.25,b=0.18,c=0.24,d=0.33],\"),u(n),reme mber);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'nuprobG:6#%\"nG6\"6#%)rem emberGE\\s3\"\"\"\"\"!\"#7$\"2+++++DiZ\"!#C\"\"&F-\"#9$\"6+++++++]t&)) !#G\"\"$F-\"\"(F-\"#6F-\"#:$\"9++++++++]Aw9!#I\"#;$\";+++++++]Q.a8A!#K \"#<$\"=++++++++]:5o(4$!#M\"\"*F-\"#8$\"4++++++v'GW!#E\"\"#F-\"\")F-\" \"'F-\"#5F-\"\"%F-F-F-,F-9!6#,&9$F,!#=F,$!:+++++++]P8Oz\"!#O-FR6#,&FUF ,!#\"FE-FR6#,&FUF,!#;F,$!9+++++++D#znx#FA-FR6#,&FUF ,!#:F,$\"8++++++++T(*\\#F=-FR6#,&FUF,!#9F,$!6++++++]?1<'F6-FR6#,&FUF,! #8F,$\"5++++++vNL5FJ-FR6#,&FUF,!#7F,$!4+++++vpA.#F1-FR6#,&FUF,!#6F,$\" 3++++++m!e#!#A-FR6#,&FUF,!#5F,$!1+++++$yv'!#?-FR6#,&FUF,!\"*F,$\"/++++ +hlFV-FR6#,&FUF,!\")F,$!.++++P'QF]o-FR6#,&FUF,!\"(F,$\"-++++H))Fio-FR6 #,&FUF,!\"'F,$!,++]kY#Fep-FR6#,&FUFW,!\"&F,$\"*+++V#Fbq-FR6#,&FUF,!\"%F ,$!'++\")F_r-FR6#,&FUF,!\"$F,$\"(+++\"F[s-FR6#,&FUF,!\"#F,$!&++$Fgs-FR 6#,&FUF,!\"\"F,$\"$+$FctF(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 207 " The probabilities for this non-uniform model are rather different from those obtained in the uniform model: the pattern abacab is now less p robable, since b and c are less probable than in the uniform model." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "for i from 0 by 500 to 100 00 do i,nuprob(i) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!F#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"$+&$\"S+\"[*))R%R^DT'*efa)Qa))oCVq*G 6m\"!#_" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+5$\"R@r=I`R]'G&p/loOnM* zKz)R0$)Q'!#^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+:$\"SN&oyL$z1h-5- X#\\]1\"H#3[k=(Qi8!#^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+?$\"Svj\" GW)>&H:u>'4#eivswgJfOuvG#!#^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+D$ \"STHpgHM!*z/s$34`+FP27m$eH`d%!#^" }}{PARA 11 "" 1 X"" {XPPMATH 20 "6$\"%+N$\"Ru%R$Ra'*pP!zcD0**oRlAjni#oq'e!#]" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$\"%+S$\"RHn*3/Y2m48;t(\\YrA4)eO^7u5wx!>9^3z#f'4VT ++\"!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+b$\"S9X8\\;$pd'oGA5')[C e&>UB6T.$R6!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+g$\"Sb5u?GjkQ%*R wF&[+z7&Q;.N>^w7!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+l$\"S3)*))) Hp*=4O+$RmNb1lD'R)R4!Q59!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+q$ \"S6^]$p&f*eM:z(Q-v$[de;hw0T)R:!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ \"%+v$\"SV=IG+`YDJqeY!Qg!z@1dF8I-k;!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+!)$\"SEB+a.Z\\AARQO&Hh+p/bIY]EAy\"!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+&)$\"S,];'QAq'p(He.)RF%yr!)3HH'R!R*=!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+!*$\"S\">B:E0Cf\\D(oz1@xDP@e4Q7k)*>!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+&*$\"SDqt/[)y\"p\"pa0<-'p(yMew\\ &49'4#!#]" }}{PARA 11 "" 1 "" {XPPYMATH 20 "6$\"&++\"$\"Ss!Hg*)RJ\\^.w5 L?GVYej!=Lv?'=#!#]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Here are th e curves corresponding to the non-uniform and to the uniform model:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "plots[display](\{plot([se q([100*i,p(100*i)],i=0..100)]),plot([seq([100*i,nuprob(100*i)],i=0..10 0)])\});" }}{PARA 13 "" 1 "" {INLPLOT 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fp7$Fb[l`$\"SU1V%*=f8CUB6T.$R6Ffp7$F[]l$\"SaGJ)[1aBs?Z,B-%fvxJYtdR&p;\"Ffp7$F`]l$\"S4=bRb6 TEj<7')\\p<*p)\\^xT0^%>\"Ffp7$Fe]l$\"Sif[Mk()GzM(R8(Re-/5290A\"Ffp 7$Fj]l$\"Sop%)RiWMTRa+j^#[%eiDw$[h*H\\7Ffp7$F_^l$\"Sb5u?GjkQ%*RwF&[+z7 &Q;.N>^w7Ffp7$Fd^l$\"S2SF\"3Amd4)e\\m2?4s\\MY[P**e.8Ffp7$Fi^l$\"Sfn&GJ j$y#fRV'3F;x&HDt,NVC0L\"Ffp7$F^_l$\"S`%R)*=i6k7I%4oh6$4hBM5vd1tN\"Ffp7 $Fc_l$\"Sh.Q7AW0w]b4)4aG)[@=L*)ey#RQ\"Ffp7$Fh_l$\"S3)*)))Hp*=4O+$RmNb1 lD'R)R4!Q59Ffp7$F]`l$\"SWC\"zh))yUh=7_VKP.Hi)*>#=alO9Ffp7$Fb`l$\"S@m@- ^Ch:O@,>!f\"Ffp7$F`bl$\"S$\\EE?FmQ?!>`tDpF;J$[>)*3K]h\"Ffp 7$Febl$\"SmpH;K6^HWlsku+dt\"Ffp7$F^d l$\"S:PF\"*Q$>!>8BV2r5>$f;7E%*R#4fFfp7$Fffl$\"Sk85B[_Fq` ;$oSG`2VfSo_z]m$>Ffp7$F[gl$\"S^f$=9&f*fKdugn)zP(y%Q/!fh+w&>Ffp7$F`gl$ \"SIKFRa)*R29IoY1H$yc_*e!y#[Ey>Ffp7$Fegl$\"S\">B:E0Cf\\D(oz1@xDP@e4Q7k )*>Ffp7$Fjgl$\"S]=<2\"oYf7)G$zXEH(=kg/&4#ys=?Ffp7$F_hl$\"S:X\"3%\\HFc \"G/$=mn'>mP?MIrA&Q?Ffp7$Fdhl$\"SFR6#[c]`.W/kB@k%)=?N2!4U-e?Ffp7$Fihl$ \"Sd=S4,BW6%QiY)R)*pjY,H8d2Bx?Ffp7$F^il$\"SDqt/[)y\"p\"pa0<-'p(yMew\\& 49'4#Ffp7$Fcil$\"S$3gG)fU?^\"o$R-VDr8_F%)>XNv9@Ffp7$Fhil$\"S6okc'Q+s-C T+0bk2q>9M9TnI8#Ffp7$F]jl$\"Su)yVA-:J4nH[#oUUs&zjkUd\"3^@Ffp7$Fbbjl$\"S AhmLNHq8r[d$Q-&e_/]?F(=&zo@Ffp7$Fgjl$\"Ss!Hg*)RJ\\^.w5L?GVYej!=Lv?'=#F fpF[[m" 2 294 214 214 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 7896 0 0 0 0 0 0 }}} {PARA 5 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 15 " Markovian model" }}{PARA 0 "" 0 "" {TEXT -1 444 "By slightly modifying the automaton, it is possible to consider a Markovian model, where in stead of giving the probabilities of occurrence of each letter, one gi ves the probabilities of transition from one letter to the next one. T he new automaton is almost the same as the previous one, except that t he transitions are marked and three more states are added at the begin ing. Here is the procedure generating the new automaton from the patte rn:" }}{EXCHG {PARA 4 "> " 0 "" {MPLTEXT 1 0 858 "gengram2 := proc (pa ttern::list(\{identical(a), identical(b),identical(c), identical(d)\}) )local i, eq, letter, state, j,alph;alph:=[a,b,c,d];for i to nops(patt ern) do for letter in aclph do for j from 0 to i-1 do if [op(1 .. i-j,p attern)] =[op(j+1 .. i-1,pattern), letter] then state[letter] :=cat(w, op(1 .. i-j,pattern));break fi od;if j=i then state[letter] := cat(w,l etter) fi od;eq[i] := cat(w,op(1 .. i-1,pattern)) =Union(Epsilon,seq(P rod(letter,`if`(i>1,cat(Mark,pattern[i-1],letter),cat(Markini,letter)) ,state[letter]), letter = alph))od;subs(cat(w,op(pattern)) =Prod(Ma rk,w),\{Mark = Epsilon,seq(seq(cat(Mark,i,j)=Epsilon,j=alph),i=alph),s eq(eq[i],i = 1 .. nops(pattern)),seq(letter = Atom,letter=alph), \+ seq(cat(w,i)=Union(Epsilon,seq(Prod(j,cat(Mark,i,j),cat(w,j)),j=alph)) ,i=subs(pattern[1]=NULL,alph)),seq(cat(Markini,i)=Epsilon,i=alph)\})en d;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)gengram2G:6#'%(patternG-%%lis tG6#<&-%*identicalG6#%\"aG-F.6#%\"bG-F.6#%\"cG-F.6#%\"dG6(%\"iG%#eqG%' letterG%&stateG%\"jG%%alphG6\"FAC%>8)7&F0F3F6F9?(8$\"\"\"FH-%%nopsG6#9 $%%trueGC$?&8&FDFMC$?(8(\"\"!FH,&FGFH!\"\"FHFM@$/7#-%#opG6$;FH,&FGFHFS FVFL7$-Fen6$;,&FSFHFHFHFUFLFPC$>&8'6#FP-%$catG6d$%\"wGFZ%&breakG@$/FSFG >F`o-Fdo6$FfoFP>&8%6#FG/-Fdo6$Ffo-Fen6$;FHFUFL-%&UnionG6$%(EpsilonG-%$ seqG6$-%%ProdG6%FP-%#ifG6%2FHFG-Fdo6%%%MarkG&FL6#FUFP-Fdo6$%(MarkiniGF PF`o/FPFD-%%subsG6$/-Fdo6$Ffo-FenFK-F_q6$FgqFfo<(/FgqFjp-F\\q6$/-Fdo6$ F\\rFGFjp/FGFD-F\\q6$/-Fdo6$FfoFG-Fhp6$Fjp-F\\q6$-F_q6%FS-Fdo6%FgqFGFS -Fdo6$FfoFS/FSFD/FG-F_r6$/&FL6#FH%%NULLGFD-F\\q6$/FP%%AtomGF]r-F\\q6$- F\\q6$/FjsFjpF^tF^s-F\\q6$F^p/FG;FHFIFAFA" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "For instance, the same pattern abacab as above yields th e automaton described by the following grammar" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "G:=gengram2([a,b,a,c,a,b]);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%\"GG-F56'F(-F86%F*%'MarkbaGFY-F 86e%F-%'MarkbbGFQ-F86%F/%'MarkbcGFB-F86%F1%'MarkbdGFF/FF-F56'F(-F86%F*% 'MarkdaGF3-F86%F-%'MarkdbGFQ-F86%F/%'MarkdcGFB-F86%F1%'MarkddGFF/FB-F5 6'F(-F86%F*%'MarkcaGF3-F86%F-%'MarkcbGFQ-F86%F/%'MarkccGFB-F86%F1%'Mar kcdGFF/FTF(/FWF(/FPF(/FMF(/Fhn-F56'F(-F86%F*F\\q%'wabacaGF]qF`qFcq/F:F (/FAF(/F=F(/F_r-F56'F(F7-F86%F-F=-F86$F'FHF?FC/FfpF(/FeqF(/F]pF(/F`pF( /FcpF(/FgoF(/F\\qF(/F_qF(/FbqF(/FEF(/FaoF(/FdoF(/F^oF(/FQ-F56'F(-F86%F *F^oF3F_oFboFeo" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "From there, g enerating functions follow. For instance, we recover the generating fu nction obtained before when all transitions are equiprobable:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "subs(gfsolve(G,unlabelled,z ,[[u,Mark],seq([1/4,cat(Markini,i)],i=[a,b,c,d]),seq(seq([1/4,cat(Mark ,i,j)],j=[a,b,c,d]),i=[a,b,c,d])]),w(z,u));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&\"$c#\"\"\"*$%\"zG\"\"%F'F',.!%'4%F'F)\"%'4%*$F) \"\"'!\"\"*$F)\"\"&\"#;*&F)F/%\"uGF'F'F(!#;F0F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "The rest of the treatfment is as before." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 17 "Multiple patterns" }}{PARA 0 "" 0 "" {TEXT -1 284 "It is also possible to writ e an automaton which will recognize not only a fixed pattern, but a se t of possible patterns. The procedure gengram3 below takes as input a \+ list of patterns, and produces the minimal grammar recognizing all wor ds on 4 letters, the patterns being ``marked''." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 913 "gengram3:=proc (patterns::list(list(\{identic al(a),identical(b),identical(c),identical(d)\}))) local alpha, states, state, eq, n, trans, letter, nbst, st, indst, i; alpha:=[a,b,c,d]; nb st:=1; states[1]:=[]; for indst while indst<=nbst do state:=states[ind st]; n:=nops(state); for letter in alpha do if member([op(state),lette r],patterns) then trans[letter]:=Prod(Mark,w) elif member([op(state),l etter],map( proc(x,n) if nops(x)>n then [op(1..n+1,x)] fi end, pattern s,nops(state))) then trans[letter]:=cat(w,op(state),letter); ngbst:=nbs t+1; states[nbst]:=[op(state),letter] else for i from indst by -1 to 2 do st:=states[i]; if st=[op(n-nops(st)+2..n,state),letter] then trans [letter]:=cat(w,op(st)); break fi od; if i=1 then trans[letter]:=w fi \+ fi od; eq[indst]:=cat(w,op(state))= Union(Epsilon,seq(Prod(letter,tran s[letter]),letter=alpha)) od; \{seq(eq[i],i=1..nbst),Mark=Epsilon,seq( letter=Atom,letter=alpha)\} end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% )gengram3G:6#'%)patternsG-%%listG6#-F*6#<&-%*identicalG6#%\"aG-F06#%\" bG-F06#%\"cG-F06#%\"dG6-%&alphaG%'statesG%&stateG%#eqG%\"nG%&transG%'l etterG%%nbstG%#stG%&indstG%\"iG6\"FHC'>8$7&F2F5F8F;>8+\"\"\">&8%6#FO7 \"?(8-FOFOFH1FVFNC&>8&&FR6#FV>8(-%%nopsG6#FZ?&8*FK%%trueG@'-%'memberG6 $7$-%#opGF[oF]o9$>&8)6#F]o-%%ProdG6$%%MarkG%\"wG-Fao6$Fco-%$mapG6%:6$% \"xGFAFHFHFH@$29%-Fjn6#Ffo7#-Feo6$;FO,&FjpFOFOFOFfoFHFHFfoFinC%>Fho-%$ catG6%F_pFdoF]o>FN,&FNFOFOFO>&FR6#FNFcoC$?(8.FV!\"\"\"\"#F^oC$>8,&FR6# F^r@$/Fcr7$-Feo6$;,(FhnFO-Fjn6#FcrF_rF`rFOFhnFZF]oC$>Fho-Feq6$F_p-FeoF ^s%&breakG@$/F^rhFO>FhoF_p>&8'Ffn/-Feq6$F_pFdo-%&UnionG6$%(EpsilonG-%$s eqG6$-F\\p6$F]oFho/F]oFK<%/F^pFat-Fct6$&FjsFer/F^r;FOFN-Fct6$/F]o%%Ato mGFgtFHFH" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "For instance, here i s the grammar recognizing the words abab and abacab:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "G:=gengram3([[a,b,a,b],[a,b,a,c,a,b]]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"GG<-/%\"wG-%&UnionG6'%(EpsilonG- %%ProdG6$%\"aG%#waG-F-6$%\"bGF'-F-6$%\"cGF'-F-6$%\"dGF'/F0-F)6'F+F,-F- 6$F3%$wabGF4F7/F?-F)6'F+-F-6$F/%%wabaGF1F4F7/%%MarkGF+/F/%%AtomG/F3FI/ F6FI/F9FI/%&wabacG-F)6'F+-F-6$F/%'wabacaGF1F4F7/FS-F)6'F+F,-F-6$F3-F-6 $FGF'F4F7/FE-F)6'F+F,FW-F-6$F6FNF7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Again, generating functions follow:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 48 "subs(gfsolve(G,unlabelled,z,[[u,Mark]]),w(z,u));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,(\"\"\"F&*$%\"zG\"\"#F&*$F(\"\"% F&F&,4!\"\"F&F(F+F'F-*$F(\"\"$F+F*!\"#*&F(F+%\"uGF&F&*$F(\"\"&F+*$F(\" \"'F-*&F(F6F2F&F&F-F-" }}}{EXCHG {PARA 0 "" i0 "" {TEXT -1 19 "The coef ficient of " }{XPPEDIT 18 0 "u^k*z^n" "*&)%\"uG%\"kG\"\"\")%\"zG%\"nGF &" }{TEXT -1 84 " in the Taylor expansion of this rational function is the number of words of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 75 " occurrences of the patterns abab and abacab. Here are the first few terms:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "map(expand,series(\",z,10)); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+9%\"zG\"\"\"\"\"!\"\"%\"\"\"\"#; \"\"#\"#k\"\"$,&\"$b#F%%\"uGF%\"\"%,&\"%;5F%F/\"\")\"\"&,&\"%[SF%F/\"# [\"\"',&\"&Gh\"F%F/\"$c#\"\"(,(\"&eU'F%F/\"%x7*$F/\"\"#F%\"\"),(\"'?gD F%F/\"%7hF@\"#7\"\"*-%\"OG6#F%\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "For instance the term " }{XPPEDIT 18 0 "48*u*z^6" "*(\"#[\"\"\" %\"uGF$%\"zG\"\"'" }{TEXT -1 364 " corresponds to the 47 words of leng th 6 containing abab (ababab is counted only once), plus the word abac ab itself. Of course, it would not be difficult to modify the grammar \+ soj as to take into accounts overlapping words differently. From this g enerating function, the treatment proceeds as before. Again, non-unifo rm and Markovian extensions could be considered." }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 20 "Patterns with errors" }}{PARA 0 "" 0 "" {TEXT -1 612 "It is also possible to accomodate patterns whose occurrence is ex act except at one unspecified position. A direct way would be to apply the previous technique for multiple patterns after having generated a ll possible patterns obtained by introducing one error in the pattern \+ under study. However, the number of patterns obtained this way may be \+ much too large for this technique to be practical. A better way is to \+ produce directly the automaton corresponding to occurrences of the pat tern with at most one error, and this turns out not to be too difficul t. The following procedure generates all words of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 148 ", exact occurrences of the pattern \+ being tagged with a mkark as before (Mark0err), while occurrences with \+ one mismatch are tagged by a Mark1err mark." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1479 "gengram4:=proc (pattern::list(\{identical(a),iden tical(b),identical(c),identical(d)\})) local alpha, state, eq, n, lett er, st, i, eq2, staterr, j; alpha:=[a,b,c,d]; for i to nops(pattern) d o staterr[i]:=\{\}; for letter in alpha do for j from 0 to i do if j=i or [op(1 .. i - j, pattern)] = [op(j + 1 .. i - 1, pattern), letter] \+ then if j=0 then state[letter] := cat(w, op(1 .. i , pattern)) else st aterr[i]:=staterr[i] union \{[[op(1..i,pattern)], [op(1..i-j,pattern)] ]\}; state[letter]:=cat(w,op(1..i,pattern),`|`, op(1..i-j,pattern)) fi ; break fi od; od; eq[i] := cat(w, op(1 .. i - 1, pattern)) = Union(Ep silon, seq(Prod(letter, state[letter]), letter = alpha)) od; for i to \+ nops(pattern)-1 do for st in staterr[i] do n:=nops(st[2]); for letter \+ in alpha do for j from 0 to n+1 do if j=n+1 or [op(1..n-j+1,pattern)]= [op(j+1..n,st[2]),letter] then if pattlern[i+1]=letter then state[lett er]:=cat(w,op(1..i+1,pattern),`|`, op(1..n-j+1,pattern)); staterr[i+1] :=staterr[i+1] union \{[[op(1..i+1,pattern)],[op(1..n-j+1,pattern)]]\} else state[letter]:=cat(w,op(1..n-j+1,pattern)) fi; break fi od od; e q2[st] := cat(w,op(st[1]),`|`,op(st[2])) = Union(Epsilon, seq(Prod(let ter, state[letter]), letter = alpha)) od od; \{seq(eq[i],i=1..nops(pat tern)), seq(seq(eq2[st],st=staterr[i]),i=1..nops(pattern)-1), cat(w,op (pattern))=Prod(Mark0err,w), seq(cat(w,op(st[1]),`|`,op(st[2]))=Prod(M ark1err,w), st=staterr[nops(pattern)]), Mark0err=Epsilon,Mark1err=Epsi lon,seq(letter=Atom,letter=alpha)\} end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)gengram4G:6#'%(patternG-%%listG6#<&-%*identicalG6#%\"aG-F.6#% \"bG-F.6#%\"cG-F.6#%\"dG6,%&alphaG%&stateG%#eqG%\"nG%'letterG%#stG%\"i G%$eq2G%(staterrG%\"jG6\"FEC&>8$7&F0F3F6F9?(8*\"\"\"FL-%%nopsG6#9$%%tr ueGC%>&8,6#FK<\"?&8(FHFQ?(8-\"\"!FLFKFQ@$5/FenFK/7#-%#opG6$;FL,&FKFLFe n!\"\"FP7$-F]o6$;,&FenFLFLFL,&FKFLFaoFLFPFYC$@%/FenFfn>&8%6#FY-%$mcatG6 $%\"wG-F]o6$;FLFKFPC$>FT-%&unionG6$FT<#7$7#FcpF[o>F\\p-F`p6&FbpFcp%\"| grGF\\o%&breakG>&8&FV/-F`p6$Fbp-F]o6$;FLFgoFP-%&UnionG6$%(EpsilonG-%$s eqG6$-%%ProdG6$FYF\\p/FYFH?(FKFLFL,&FMFLFaoFLFQ?&8)FTFQC%>8'-FN6#&Fjr6 #\"\"#?&FYFHFQ?(FenFfnFL,&F]sFLFLFLFQ@$5/FenFes/7#-F]o6$;FL,(F]sFLFenF aoFLFLFP7$-F]o6$;FfoF]sF`sFYC$@%/&FP6#,&FKFLFLFLFYC$>F\\p-F`p6&Fbp-F]o 6$;FLFhtFPFaqF[t>&FUFgt-Fip6$Fau<#7$7#F]uFjs>F\\p-F`p6$FbpF[tFbq>&8+6# Fjr/-F`p6&Fbp-F]o6#&Fjr6#FLFaq-F]oF_sF\\r<)/%)Mark0errGF_r/%)Mark1errG F_r-Far6$/FY%%AtomGFfr-Far6$/F_v-Fdr6$FjvFbp/Fjr&FU6#FM/-F`p6$Fbp-F]oF O-Fdr6$FhvFbp-Far6$-Far6$F[v/FjrFT/FK;FLFhr-Far6$Fdq/FK;FLFMFEFE" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "Here is a simple example. The gen erated automaton is almost optimal (in general it has O(1) too many st ates):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "gengram4([a,b,b,c ]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<5/%\"aG%%AtomG/%\"bGF&/%\"cGF& /%\"dGF&/%#waG-%&UnionG6'%(EpsilonG-%%ProdG6$F%%&wab|graG-F46$F(%$wabG -F46$F*%%wab|grG-F46$nF,F " 0 "" {MPLTEXT 1 0 67 "subs(gfsolve(\"o,unlabelled,z,[[u,Ma rk0err],[v,Mark1err]]),w(z,u,v));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#, $*&,4!\"\"\"\"\"%\"zG!\"%*$F(\"\"#!\"'*$F(\"\"$F,*$F(\"\"%F)*$F(\"\"&! \"$*$F(\"\")F.*$F(\"\"'!\"#*$F(\"\"*F.F',L*&F(F2%\"uGF'F)*&F(\"\"(F=F' !\"&F'F'F1\"#@F-!#=*&F(F2%\"vGF'!#I*&F(F7F=F'F,*&F(F7FDF'!#@F4F3F9F3F6 \"#<*$F(F?F.*&F(F?FDF'F,*&F(F:FDF'F:*&F(\"#5FDF'F:*&F(FNF=F'F.*&F(F:F= F'F.*&F(F0F=F'F&F*!#5*&F(F0FDF'!#7F/!\"(F&F&" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 19 "The coefficient of " }{XPPEDIT 18 0 "u^k*v^l*z^n" "*()% \"uG%\"kG\"\"\")%\"vG%\"lGF&)%\"zG%\"nGF&" }{TEXT -1 54 " in the Taylo r expansion of this rational function at " }{XPPEDIT 18 0 "z=0" "/%\"z G\"\"!" }{TEXT -1 34 " is the number or words of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 31 " where the pattern abbc occurs " } {XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 25 " times without error and " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 149 " times with exactly one e rror. Again, the Markovian model could also be treated, and extensions to multipple errors are likely to be possible as well." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Conclusion" }}{PARA 0 "" 0 "" {TEXT -1 575 "Various proba bilistic parameters related to the occurrence of specific patterns in \+ random words can be computed very easily using combstruct and gfun. He re, combstruct is used to model the combinatorics of the problem and g fun is very helpful to compute expansions to very large orders, thanks to the rational type of the corresponding generating functions. The m odel itself can be modified in various directions, to take into accoun t different probability models or different ways of counting occurrenc es of the pattern when they overlap, or several patterns simultaneousl y." }}}}{MARK "5 0 0" 46 }{VIEWOPTS 0 0 0 1 1 1803 } \"kG6\"" }{TEXT -1 25 " times without error and " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 149 " times with exactly one e rror. Again, the Markovian model could also be treated, and extensions to multiporderingordinarC}ordinatore3 !#' 1cfry]org.TmEorganizxorgano+orixorient }]orientatorigin;]fyxoriginal3 !koriginatx orthogonalWorthopo +Kother9  +.!>?A9Da%fmFs>ԡ}xEjotherwisG?+KTb!kl7pyĦ׵½ouput our?+ 1]y}xEancgraphxangerjWangew}anneyannouncy}W]annual  janonymou!k anonymous!kanoth'yxjansi q<answany{F+ !/!>9D+K]bf!kryy}xEanythQiap } apart 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2 3 "DEC ALPHA UNIX" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "courier" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Ou tput" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3 " 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0y -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Error" 7 8 1 {CSTYLE "" -1 -1 "" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 z"" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 256 44 "Combinatorics of Non-Crossing Configurations" }}{PARA 257 "" 0 "" {TEXT -1 22 "F. Cazals, August 1997" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Generalities" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 27 "Non-Crossing configurations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Take " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 0 "" } {TEXT -1 178 " points equally spaced on the unit circle, and draw chor ds between these points with the constraint that no two chords cross o ne-another. The resulting configuration is called a " }{TEXT 257 27 "n on-crossing configuration " }{TEXT -1 611 "and the study of such entit ies originates in the work of Euler and Segner in 1753 for counting tr iangulations of a n-gon. Since then several types of such configuratio ns have been defined, and for example the presence/absence of cycles \+ and the number of connected components in a given configuration define the classes of trees and forests, connect{ed graphs and general graphs . In addition to be of combinatorial interest per se, these configura tions are also important for algorithmic problems arising in computer \+ graphics or computational geometry where they provide simple models fo r real-world situations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 446 "If historically the study of these configurations has been carried out one at a time, it turns out that they all fit in the model of algebraic and analytic combinatorics and are thus amenab le to a unified treatment. More precisely, they can be defined in ter ms of grammars, from which the generating functions can automatically \+ be obtained and used to asymptotically analyse the number of configura tions, the number of connected components, etc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 339 "The goal of this workshe et is to present this chain, from the grammars specification using com bstruct, to the asymptotics of algebraic functions using some featur|es of gfun, together with some plots of random configurations and of the singularities determining the asymptotic behaviour. The reader is ref erred to [FlaNo97] for the details." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 525 "The worksheet is organized as follows. \+ In the rest of this section we define a few functions of null combina torial interest but that we shall need to plot random configurations. \+ In section two we present the grammar specifications for 6 of the main non-crossing configurations together with some plots of random config urations. And section three is devoted to the asymptotic machinery use d to count the number of configurations of a given size, as well as a \+ comparison between the asymptotic estimates and the exact values. " }} {PARA 0 "" 0 "" {TEXT -1 75 "\nBut to begin with, we first load the co mbstruct, gfun and plots libraries:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "with(combstruct):with(gfun):with(plots):" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT} -1 8 "Appendix" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 154 "Again, we define here a few functions we shall need to p lot NC configurations. The first one returns the number of atoms in a \+ structure, that is its size:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "size:=proc(t) convert(map(size, t), `+`) end:size(Epsilon):=0: s ize(Z):=1:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 203 "Since a configurat ion is defined by edges drawn between points equally spaced on the uni t circle, we first show how to retrieve these pairs of indices from th e grammars to be defined in the next section:" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 33 "Collecting the edges of a NC-tree" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "Since a tree T satisfies T=Product (Z, Sequence( Butterflies)), to plot it we just have to collect its edges which are \+ pairs of indices in 0..n-1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 307 "plotTree:= proc(aTree) local r, nbVertices;\n #--op returns the S equence; 0 is the index of the root on the circle;\n~ #--1 indicates th at the butterflies attached to the root are counted as a \n #--right w ing \n r:=getTreeEdges(op(2, aTree), 0, 1);\n nbVertices:=size(aTree); \n\n plotSetOfEdges(r, nbVertices)\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 270 "The parameters of getTreeEdges are the following: aSeqO fBtf is a sequence of butterflies; rootIdx is the index of the vertex this sequence is attached to; leftRight (-1 or +1) indicates if this sequence is a left/right wing of the bug. The algorithm works as foll ows:" }}{PARA 0 "" 0 "" {TEXT -1 70 "1.We first compute the indices of the apex vertices of the butterflies" }}{PARA 0 "" 0 "" {TEXT -1 95 " 2.For a given butterfly, we recurse on the 2 wings and attach its apex to the rootIdx parameter" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 985 "getTreeEdges:=proc(aSeqOfBtf, rootIdx, leftRight) \n local i, a, \+ eL, currentBtf, cumul;\n \n if aSeqOfBtf = Epsilon then \{\}\n else\n \+ cumul := rootIdx;\n if leftRight=1 then #--we are processing a right wing\n for i to nops(aSeqOfBtf) do \n currentBtf := op(i, aSeqOf Btf);\n a[i] := cumul + size(op(1, currentBtf)) + 1;\n cumul := cumul + size(currentBtf)\n od; \n else #--and here \+ a left one \n for i from nops(aSeqOfBtf) by -1 to 1 do\n \+ currentBtf := op(i, aSeqOfBtf);\n a[i] := cumul - size(op(3, cur rentBtf)) - 1;\n cumul := cumul - size(currentBtf);\n od;\n fi; \n \n #-- recurses for each Prod(?, Z, ?) in the sequence, wi th ? = E or Seq()\n eL := \{\};\n for i from 1 to nops(aSeqOfBtf) do \n currentBtf := op(i, aSeqOfBtf);\n eL := eL union getTreeEdge s(op(1, currentBtf), a[i], -1);\n eL := eL union \{[a[i], rootIdx] \};\n eL := eL union getTreeEdges(op(3, currentBtf), a[i], 1);\n o d;\n \n #--returns the result\n eL;\n fi;\nend:" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 34 "Collecting the edges of a NC-graph" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 199 "This first procedure recursively collect s the edges of an EA, and is a straightforward application of the EA d efinition above. The parameter ori stands for the index of the lefmost point of the arch:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 595 "get ArchEdges:=proc(arch, ori) local i, offset, res;\n\nif (arch=Epsilon) \+ then res:=\{\}\nelse\n #--we first add the `roof' of the arch\n res := \{[ori, ori+size(arch)]\};\n \n #-----Prod(Z, Seq1, Seq2)\n if (op(1, arch)=Z) then \n res := res union getArchEdges(op(2,ar ch), ori);\n res := res union getArchEdges(op(3,arch), ori+1+size(op( 2,arch)))\n else #--Sequence of arches\n offset:=0;\n #--let's proce ss all the arches in this sequence \n for i from 1 to nops(arch) do\n res := res union getArchEdges(op(i, arch), ori+offset);\n offset \+ := offset + size(op(i,arch))\n od\n fi\nfi;\nres;\nend:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Same thing but to a sequence of arches:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 309 "getArchesSeqEdges:=proc( archesSeq, ori) \n local i, arch, offset, res;\n res:=\{\}; offset:=0; \n \n for i from 1 to nops(archesSeq) do\n arch:= op(i, archesSeq);\n res:= res union getArchEdges(arch, ori+offset);\n offset := offset+ size(arch)\n od;\n #--returns the setOfEdges and the new origin\n res ,ori+offset;\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "And now the main procedure which collects the edges of a Non-Crossing graph:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1102 "getNCGraphEdges:=proc(aNCG raph) \n local res, ori, edges, \n i, j,\n seqOfSeqOrProd, seqOrProd;\n \n #--first, the right ear which is a Seq(EA)\n ori:=1; \n res:=getArchesSeqEdges(op(3,aNCGraph), ori);\n edges:=res[1]; ori : = res[2];\n\n edges := edges union \{[0, ori]\};\n \n #--then the EAs inbetween two successive childs of v_1\n seqOfSeqOrProd := op(5,aNCGr aph);\n \n for i from 1 to nops(seqOfSeqOrProd) do\n seqOrProd:= op(i, seqOfSeqOrProd);\n #--2 connected graphs\n if (op(1, seqOrPr od)=Z) then\n res:=getArchesSeqEdges(op(2, seqOrProd), ori);\n e dges:= edges union res[1]; ori := res[2];\n \n res:=getArchesSeqE dges(op(3, seqOrProd), 1+ori);\n edges:= edges union res[1]; ori := res[2]\n\n #--a Sequence\n else \n res :=getArchesSeqEdges(seqOrProd, ori);\n edges:= edges union res[1]; \+ ori := res[2]\n fi;\n\n #--we need to add the current child to the graph root\n edges := edges union \{[0, ori]\};\n od;\n\n #--and th e left ear\n res:=getArchesSeqEdges(op(4,aNCGraph), ori);\n edges:= ed ges union res[1]; ori := res[2];\n\n #--returns the result\n edges\nen d:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "To plot a NC Graph, we just collect the edges and pass them to the plotSetOfEdges procedure:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "plotNCGraph:=proc(aNCGraph); \n plotSetOfEdges(getNCGraphEdges(aNCGraph), size(aNCGraph))\nend:" }} }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 228 "Now, given pairs of indices in \+ 0,...,nbVertices-1 on the unit circle, the following procedure draws t he corresponding chords\nassuming that the k-th point has coordinates \+ ((cos((2 Pi k)/nbVertices), sin((2 Pi k)/nbVertices)) ):\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 208 "plotSetOfEdges:=proc(aSetOf Edges, nbVertices) local pointsOnCircle;\n\n pointsOnCircle:= expand(m ap(`*`, aSetOfEdges,2*Pi/nbVertices));\n plot([op(map2(map,[cos,sin], \+ pointsOnCircle))],color=blue,axes=NONE)\nend:" }}}}}{SECT 0 {PARA 3 " " 0 "" {TEXT -1 48 "Counting and drawing non-crossing configurations" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 17 "Trees and forests" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 5 "Trees" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 470 "A Tree is a sequence of butterfli es attached to a root, a Butterfly being an ordered pair of trees whos e roots have been merged into a single node. Since this merge step ca nnot be specified by an operation such as B = Prod(T, Z, T)/Z in combs truct, it is more convenient to express a butterfly as the product of \+ 2 forests, a Forest being a sequence of butterflies. Indeed, we now j ust have to attach the roots of all the trees of the two forests to a \+ newly added node:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "tbf:= \{T=Prod(Z, Sequence(B)), F=Sequence(B), B=Prod(F,Z,F)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tbfG<%/%\"TG-%%ProdG6$%\"ZG-%)SequenceG6#%\" BG/%\"FGF,/F/-F)6%F1F+F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Stand ard functionalities of Combstruct consist in counting the number of en tities of a given type:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " seq(count([T,tbf], size=i),i=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6,\"\"\"F#\"\"$\"#7\"#b\"$t#\"%G9\"%_x\"&jK%\"'vmC" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "seq(count([F,tbf], size=i),i=1..10);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"\"\"\"$\"#7\"#b\"$t#\"%G9\"%_x\"&j K%\"'vmC\"(:2V\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "seq(cou nt([F,\{F=Sequence(B), B=Prod(F,Z,F)\}],size=i),i=0..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-\"\"\"F#\"\"$\"#7\"#b\"$t#\"%G9\"%_x\"&jK%\"'vm C\"(:2V\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "And randomly generat ing all the configurations of a given size:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "allstructs([T,tbf],size=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#-%%ProdG6$%\"ZG-%)SequenceG6#-F%6%%(EpsilonGF'F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "expleT:=draw([T, tbf], size= 20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'expleTG-%%ProdG6$%\"ZG-%)Se quenceG6#-F&6%-F*6#-F&6%%(EpsilonGF(-F*6#-F&6%-F*6#-F&6%F2F(-F*6#-F&6% -F*6$-F&6%-F*6#-F&6%F2F(-F*6#-F&6%F2F(-F*6$-F&6%F2F(-F*6#-F&6%-F*6#-F& 6%F2F(-F*6#-F&6%F2F(F2F(F2-F&6%-F*6$FYFYF(F2F(F2-F&6%FSF(FWF(F2F(F2F(F 2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "We can also use gfseries to \+ retrieve the first terms of the generating functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Order:=10:gfseries(tbf,unlabelled,z );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7&/-%\"FG6#%\"zG+9F+ \"\"\"\"\"!F-\"\"\"\"\"$\"\"#\"#7\"\"$\"#b\"\"%\"$t#\"\"&\"%G9\"\"'\"% _x\"\"(\"&jK%\"\")\"'vmC\"\"*-%\"OG6#F-\"#5/-%\"BGF*+7F+F-\"\"\"\"\"# \"\"#\"\"(\"\"$\"#I\"\"%\"$V\"\"\"&\"$G(\"\"'\"%wQ\"\"(\"&=8#\"\")\"'v ,7\"\"*F@\"#5/-%\"TGF*+7F+F-\"\"\"F-\"\"#F0\"\"$F2\"\"%F4\"\"&F6\"\"'F 8\"\"(F:\"\")F<\"\"*F@\"#5/-%\"ZGF*+%F+F-\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "And here is an example of random tree:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plotTree(draw([T,tbf],size=20));" } }{PARA 13 "" 1 "" {INLPLOT "67-%'CURVESG6#7$7$$!3`P:&H;l0^*!#=$!3^u%\\ P%*p,4$F*7$$!3^u%\\P%*p,4)F*$!3PJZ#H__y(eF*-F$6#7$7$F($\"3^u%\\P%*p,4$ F*7$F.$\"3PJZ#H__y(eF*-F$6#7$7$F6$\"3`P:&H;l0^*F*7$F+F?-F$6#7$F-7$F0F. -F$6#7$FEF5-F$6#7$7$F?F+F5-F$6#7$7$$\"3^u%\\P%*p,4)F*F0FL-F$6#7$7$F+F( FL-F$6#7$7$F6F(FV-F$6#7$7$F9F.FP-F$6#7$7$\"\"!$!\"\"F]oFV-F$6#7$7$F]o$ \"\"\"F]oF>-F$6#7$FA7$F0FQ-F$6#7$FioF8-F$6#7$7$FQF97$FdoF]o-F$6#7$7$F? F6Fap-F$6#7$7$F^oF]oF'-F$6#7$7$F9FQF`p-F$6#7$F8Fap-%*AXESSTYLEG6#%%NON EG-%'COLOURG6&%$RGBGF]oF]o$\"*++++\"!\")" 2 263 274 274 2 0 1 0 2 6 0 1 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 255 0 0 255 1 0 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "Forests" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "In order to dissociate the trees in a fo rest, let us substitute to each vertex of a tree another vertex togeth er with a forest" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "fo:=\{ B=Prod(Sequence(B),V,Sequence(B)),\n V=Union(Prod(Z, F)),\n F= Union(Epsilon, Prod(V, Sequence(B)))\}; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#foG<%/%\"BG-%%ProdG6%-%)SequenceG6#F'%\"VGF+/F.-%&UnionG6#-F) 6$%\"ZG%\"FG/F6-F16$%(EpsilonG-F)6$F.F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "seq(count([F, fo], size=i), i=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"\"\"\"#\"\"(\"#L\"$\"=\"%$3\"\"%ao\"&6^%\"'HcI \"($G<@" }}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 28 "Connected and gener al graphs" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 16 "Connected graphs" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "To see how nc-graphs are built, co nsider the set of childs " }{XPPEDIT 18 0 "v[i],v[i+1],`...`,v[j]" "6& &%\"vG6#%\"iG&F$6#,&F&\"\"\"\"\"\"F*%$...G&F$6#%\"jG" }{TEXT -1 22 " a ttached to the root " }{XPPEDIT 18 0 "v[1]" "&%\"vG6#\"\"\"" }{TEXT -1 22 ". The graphs built on " }{XPPEDIT 18 0 "v[2],`...`,v[i]" "6%&% \"vG6#\"\"#%$...G&F$6#%\"iG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v[j], `...`,v[n-1]" "6%&%\"vG6#%\"jG%$...G&F$6#,&%\"nG\"\"\"\"\"\"!\"\"" } {TEXT -1 0 "" }{TEXT -1 67 " are connected by hypothesis, while betwee n any two other vertices " }{XPPEDIT 18 0 "v[k]" "&%\"vG6#%\"kG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "v[k+1]" "&%\"vG6#,&%\"kG\"\"\"\"\" \"F'" }{TEXT -1 67 " one can have either one connected graph or two co nnected graphs. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 254 "But any graph built from a sequence of successive vertic es of the circle is a system of arches since the arcs which are chords are not allowed to cross. The endpoints of these arches are shared by the graphs built to the left and to the right of a given " }{XPPEDIT 18 0 "v[k]" "&%\"vG6#%\"kG" }{TEXT -1 57 ", so that we shall say that \+ the size of an arch built on " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 0 "" }{TEXT -1 11 " points is " }{XPPEDIT 18 0 "n-1" ",&%\"nG\"\"\" \"\"\"!\"\"" }{TEXT -1 186 ". At last, we are interested here in Eleme ntary Arches, that is arches that always contain an arc between the fi rt and last points. General arches are easily obtained by sequencing E As. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 165 " From this discussion we derive the Combstruct specification of NC grap hs containing at least 2 vertices. In particular, the 5 arguments of a C entity are as follows:" }}{PARA 0 "" 0 "" {TEXT -1 32 "1.first Z: t he root of the graph" }}{PARA 0 "" 0 "" {TEXT -1 45 "2.second Z: the l efmost point of the first EA" }}{PARA 0 "" 0 "" {TEXT -1 30 "3 and 4. \+ the two EAs built on " }{XPPEDIT 18 0 "v[2],`...`,v[i]" "6%&%\"vG6#\" \"#%$...G&F$6#%\"iG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v[j],`...`,v[ n-1]" "6%&%\"vG6#%\"jG%$...G&F$6#,&%\"nG\"\"\"\"\"\"!\"\"" }}{PARA 0 " " 0 "" {TEXT -1 62 "5.the sequence of EAs found between two consecutiv e childs of " }{XPPEDIT 18 0 "v[1]" "&%\"vG6#\"\"\"" }{TEXT -1 0 "" } {TEXT -1 39 ". The first term corresponds to one EA," }}{PARA 0 "" 0 " " {TEXT -1 116 "and the second one to two EAs. For the latter case, th e Z in the Prod stands for the lefmost point of the secong EA." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 299 "ar:=\{EA = Union(Sequence(E A, card >= 2), \n Prod(Z, Sequence(EA), Sequence(EA))\n \+ ),\n C=Union(Z,\n Prod(Z,Z,Sequence(EA), \+ Sequence(EA),\n Sequence(Union(Sequence(EA,card>=1), \+ Prod(Z,Sequence(EA),Sequence(EA))))))\}; \n\n \+ " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#arG<$/%#EAG-%&UnionG6$-%)Seque nceG6$F'1\"\"#%%cardG-%%ProdG6%%\"ZG-F,6#F'F5/%\"CG-F)6$F4-F26'F4F4F5F 5-F,6#-F)6$-F,6$F'1\"\"\"F0F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "We can now count the number of elementary arches of a given size. The sequence found is not in [Sloa95]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "seq(count([EA,ar], size=i),i=1..20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "66\"\"\"\"\"$\"#;\"$0\"\"$o(\"%1g\"&_\"\\\"',dT\"(![ /O\")5/(=$\"*[7E'G\"+!p\"o/E\",[W'y&R#\"-+Vu*RA#\".7\\l643#\"/X+YPXg> \"0[o7qJ\"e=\"1q(GX\")e0x\"\"2+G&Q#pP^p\"\"3qYnU#)f%)H;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "allstructs([EA,ar],size=2);" }} {PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%-%) SequenceG6$-%%ProdG6%%\"ZG%(EpsilonGF+F'-F(6%F*-F%6#F'F+-F(6%F*F+F." } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "We can also count the number of NC-graphs. The corresponding sequence turns out to be M3594 in [Sloa9 5] and gives the reverse " }}{PARA 0 "" 0 "" {TEXT -1 24 "of the g.f. \+ for squares:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "seq(count([ C,ar], size=i),i=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"\"F#\" \"%\"#B\"$c\"\"%i6\"%#>*\"&>e(\"'3\\k\"(#=;c" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 83 "As usual, we can get all the structures of a given size , or just draw some of them:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "allstructs([C,ar],size=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7& -%%ProdG6'%\"ZGF'%(EpsilonGF(-%)SequenceG6#-F*6#-F%6%F'F(F(-F%6'F'F'F( F(F,-F%6'F'F'F(F,F(-F%6'F'F'F,F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "draw([EA,ar],size=10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%)SequenceG6$-F$6%-%%ProdG6%%\"ZG%(EpsilonGF,-F)6%F+F,-F$6#-F)6 %F+F,-F$6#-F)6%F+F,-F$6#F(F(-F$6$-F$6%F(F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "And an example of random graph:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 34 "plotNCGraph(draw([C,ar],size=10));" }}{PARA 13 "" 1 "" {INLPLOT "61-%'CURVESG6#7$7$$!3^u%\\P%*p,4$!#=$\"3`P:&H;l0^ *F*7$$!3^u%\\P%*p,4)F*$\"3PJZ#H__y(eF*-F$6#7$F-7$$\"3^u%\\P%*p,4)F*$!3 PJZ#H__y(eF*-F$6#7$7$$\"3^u%\\P%*p,4$F*$!3`P:&H;l0^*F*F5-F$6#7$F-7$F.F 8-F$6#7$7$$!\"\"\"\"!FLFE-F$6#7$F-7$F(F@-F$6#7$7$F6F0F5-F$6#7$F-FI-F$6 #7$FPF=-F$6#7$FT7$F>F+-F$6#7$FhnF'-F$6#7$F'F5-F$6#7$7$$\"\"\"FLFLF5-%* AXESSTYLEG6#%%NONEG-%'COLOURG6&%$RGBGFLFL$\"*++++\"!\")" 2 274 286 286 2 0 1 0 2 6 0 1 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 255 0 0 255 1 0 0 0 0 0 -6016 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 14 "Gen eral graphs" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 343 "As observed in [Fl aNo97], a general graph is obtained from a connected one by the substi tution Z -> Prod(Z, G). So that we just have to rewrite the previous \+ grammar by adding a new symbol which makes this substitution. Notice h owever that the decomposition of a connected graph misses a configurat ion for general graphs: the one where vertex " }{XPPEDIT 18 0 "v[1]" " &%\"vG6#\"\"\"" }{TEXT -1 49 " does not have any child, which we there fore add:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 384 "br:=\{EA = Un ion(Sequence(EA, card >= 2), \n Prod(V, Sequence(EA), S equence(EA))\n ),\n V=Union(Prod(Z, G)),\n G=Uni on(Epsilon,\n  Prod(Z, G), \n Prod(V,V,Sequence (EA), Sequence(EA),\n Sequence(Union(Sequence(EA,card >=1), Prod(V,Sequence(EA),Sequence (EA)))))\n )\n \};" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% #brG<%/%#EAG-%&UnionG6$-%)SequenceG6$F'1\"\"#%%cardG-%%ProdG6%%\"VG-F, 6#F'F5/F4-F)6#-F26$%\"ZG%\"GG/F=-F)6%%(EpsilonGF:-F26'F4F4F5F5-F,6#-F) 6$-F,6$F'1\"\"\"F0F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "The numbe r of graphs is given by the following sequence, not to be found in [Sl oa95]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ggSeq:=[seq(count ([G, br], size=i), i=0..20)];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&gg SeqG77\"\"\"F&\"\"#\"\")\"#[\"$_$\"%!)G\"&;_#\"'o6B\"([3>#\")K?H@\"*_V /6#\"+kkCD@\",;[&>o@\"-'ppIiB#\".kW<=yK#\"/%=hC%[UC\"0o$)p_Z0e#\"1K)oP GkHu#\"2#fiChCCJH\"3)oarGr*RZJ" }}}{EXCHG {PARA 12 "" 1 "" {TEXT -1 75 "In this case, it also turns out that the differential equation ver ified by " }{XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 15 " is of order 1: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "ggDiffEq:=listtodiffeq( ggSeq,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)ggDiffEqG7$<$/-%\"y G6#\"\"!\"\"\",*!\"\"F,%\"xG\"#=*&,(F,F,F/!#=*$F/\"\"#\"\")F,-F)6#F/F, F,*&,(F4\"#7*$F/\"\"$!\"%F/F.F,-%%diffG6$F7F/F,F,%$ogfG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "From this equation and the condition " } {XPPEDIT 18 0 "coeff(y(x), x, 1)=1" "/-%&coeffG6%-%\"yG6#%\"xGF)\"\"\" \"\"\"" }{TEXT -1 35 ", we get the following closed form:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "closedForm:=dsolve(ggDiffEq[1],y(x) );\nsubs(_C1=-1/2,closedForm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+c losedFormG/-%\"yG6#%\"xG,*\"\"\"F+F)#\"\"$\"\"#*$F)F.!\"\"*(F)F+,(F)!# 7F/\"\"%F+F+#F+F.%$_C1GF+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG 6#%\"xG,*\"\"\"F)F'#\"\"$\"\"#*$F'F,!\"\"*&F'F),(F'!#7F-\"\"%F)F)#F)F, #F.F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "As shown in [FlaNo97], t his corresponds to the general term:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "cn:=proc(n)\nlocal k,l; \nsum((-1)^k*product(2*l-1,l =1..n-k-1)/(factorial(k)*factorial(n-2*k))*3^(n-2*k)*2^(-k-2), k=0..iq uo(n,2))\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "gn:=proc( n) 2^n*cn(n-1) end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#gnG:6#%\"nG6 \"F(F(*&)\"\"#9$\"\"\"-%#cnG6#,&F,F-!\"\"F-F-F(F(" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 20 "seq(gn(i), i=3..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*\"\")\"#[\"$_$\"%!)G\"&;_#\"'o6B\"([3>#\")K?H@" }}}}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 26 "Dissections and partitions" }} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 11 "Dissections" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "A dissection of a convex polygon " }{XPPEDIT 18 0 "P [n]=\{v[1],`...`,v[n]" "/&%\"PG6#%\"nG<%&%\"vG6#\"\"\"%$...G&F)6#F&" } {TEXT -1 0 "" }{TEXT -1 135 " is a partition of the polygon into polyg onal regions by means of non-crossing diagonals. If the polygonal reg ion containing the edge " }{XPPEDIT 18 0 "v[1]*v[2]" "*&&%\"vG6#\"\"\" \"\"\"&F$6#\"\"#F'" }{TEXT -1 0 "" }{TEXT -1 5 " has " }{XPPEDIT 18 0 "r+1" ",&%\"rG\"\"\"\"\"\"F$" }{TEXT -1 54 " sides, one gets a bigger \+ dissection by replacing the " }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 0 "" }{TEXT -1 492 " edges by a dissection. So that a dissection is ei ther an edge connecting two vertices or a sequence of dissections. The tricky point in sequencing 2 dissections consists in not counting the same vertex twice, as for the connected graphs above. When sequencing dissections, we therefore assume that each dissection provides its ri ghtmost point while the lefmost one is the rightmost point of the diss ection to the left in the sequence. With this convention, the number o f dissections of size " }{XPPEDIT 18 0 "i" "I\"iG6\"" }{TEXT -1 0 "" } {TEXT -1 61 " actually counts the number of dissections of a polygon w ith " }{XPPEDIT 18 0 "i+1" ",&%\"iG\"\"\"\"\"\"F$" }{TEXT -1 0 "" } {TEXT -1 30 " vertices, and the grammar is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "dissG:=\{Di=Union(Z, Sequence(Di, card >= 2))\};\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&dissGG<#/%#DiG-%&Un ionG6$%\"ZG-%)SequenceG6$F'1\"\"#%%cardG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "The corresponding sequence, M2898 in [Sloa95] and related to Schroeder's second problem is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "seq(count([Di, dissG], size=i), i=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"\"F#\"\"$\"#6\"#X\"$(>\"$.*\"%zU\"&$z?\" '\\I5" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 11 "Partitions " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "A non-crossing partition of size " } {XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 0 "" }{TEXT -1 48 " is a partit ion of [n]=\{1,2,...,n\} such that if " }{TEXT -1 29 "a " 0 "" {MPLTEXT 1 0 38 "partG:=\{P=Sequence(V), V = Prod(Z,P)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&partGG<$/%\"PG-%)SequenceG6#%\"VG/F+-%%ProdG 6$%\"ZGF'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "We get the sequence \+ of Catalan numbers, which can be checked in terms of generating functi ons:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "seq(count([P, partG ], size=i), i=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"\"\"\"#\" \"&\"#9\"#U\"$K\"\"$H%\"%I9\"%i[\"&'z;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "gfsolve(partG, unlabelled, z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%\"ZG6#%\"zGF(/-%\"VGF',&#\"\"\"\"\"#F.*$,&F.F.F(! \"%F-#!\"\"F//-%\"PGF',$*&F(F4,&F.F.F0F4F.F-" }}}}}}{SECT 0 {PARA 3 " " 0 "" {TEXT -1 22 "Univariate asymptotics" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 30 "Asymptotic counting: algorithm" }}{SECT 0 {PARA 5 "" 0 " " {TEXT -1 7 "Outline" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "As shown \+ in [FlaNo97] for the six NC configurations we are interested in, the g enerating function " }{XPPEDIT 18 0 "y(z)" "-%\"yG6#%\"zG" }{TEXT -1 81 " satisfies an algebraic equation. In the case of NC forests for ex ample, we have:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eq:=y^3+ (-z+z^2-3)*y^2+(z+3)*y-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG,** $%\"yG\"\"$\"\"\"*&,(%\"zG!\"\"*$F,\"\"#F)!\"$F)F)F'F/F)*&,&F,F)F(F)F) F'F)F)F-F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Like in many implic itely defined functions [Dr97, HaPa73], we expect " }{TEXT 258 9 "a pr iori " }{XPPEDIT 18 0 "y(z) " "-%\"yG6#%\"zG" }{TEXT -1 63 " to have l ocally an expansion of the square-root type, that is:" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "y(z)=c[0]+c[1]*sqrt(1-z/rho) +c[2]*(1-z/rho)+O((1- z/rho)^(3/2))" "/-%\"yG6#%\"zG,*&%\"cG6#\"\"!\"\"\"*&&F)6#\"\"\"F,-%%s qrtG6#,&\"\"\"F,*&F&F,%$rhoG!\"\"F8F,F,*&&F)6#\"\"#F,,&\"\"\"F,*&F&F,F 7F8F8F,F,-%\"OG6#),&\"\"\"F,*&F&F,F7F8F8*&\"\"$F,\"\"#F8F," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "By a singularity \+ analysis at the dominant singularity " }{XPPEDIT 18 0 "rho" "I$rhoG6\" " }{TEXT -1 14 " and denoting " }{XPPEDIT 18 0 "gamma=-c[1]/2" "/%&gam maG,$*&&%\"cG6#\"\"\"\"\"\"\"\"#!\"\"F," }{TEXT -1 9 ", we get:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "[z^n] *y(z) = c[1]/\\Gamma(-1/2)* rho^(-n)/sqrt(n^3) * (1+O(1/n)) " "/*&7#)% \"zG%\"nG\"\"\"-%\"yG6#F&F(*,&%\"cG6#\"\"\"F(-%&GammaG6#,$*&\"\"\"F(\" \"#!\"\"F8F8)%$rhoG,$F'F8F(-%%sqrtG6#*$F'\"\"$F8,&\"\"\"F(-%\"OG6#*&\" \"\"F(F'F8F(F(" }{TEXT -1 5 " or " }{XPPEDIT 18 0 "y[n]=gamma*rho^(-n )/sqrt(Pi*n^3)*(1+O(1/n))" "/&%\"yG6#%\"nG**%&gammaG\"\"\")%$rhoG,$F&! \"\"F)-%%sqrtG6#*&%#PiGF)*$F&\"\"$F)F-,&\"\"\"F)-%\"OG6#*&\"\"\"F)F&F- F)F)" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "We now e xamplify this method for the class of NC-forests. The singulatities so ught may arise at those points " }{XPPEDIT 18 0 "z" "I\"zG6\"" }{TEXT -1 24 " such that E(z,y)=0 and " }{XPPEDIT 18 0 "diff(E(z,y),y)=0" "/- %%diffG6$-%\"EG6$%\"zG%\"yGF*\"\"!" }{TEXT -1 33 ". In order to get th e candidates " }{XPPEDIT 18 0 "z" "I\"zG6\"" }{TEXT -1 57 " satisfynin g these conditions, we just have to eliminate " }{XPPEDIT 18 0 "y" "I \"yG6\"" }{TEXT -1 68 " between the two previous equations through a r esultant computation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "r es_y:=resultant(eq,diff(eq,y),y);\nres_y:=expand(normal(res_y/gcd(res_ y,diff(res_y,z))));\nOmega[0]:=normal(res_y/z^ldegree(res_y));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&res_yG,**$%\"zG\"\"$!\"%*$F'\"\"%\" #K*$F'\"\"&\"\")*$F'\"\"'!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&re s_yG,*%\"zG!\"%*$F&\"\"#\"#K*$F&\"\"$\"\")*$F&\"\"%!\"&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%&OmegaG6#\"\"!,*!\"%\"\"\"%\"zG\"#K*$F+\"\"# \"\")*$F+\"\"$!\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 354 "And the si ngularities sought are the solutions of the previous equation whose mo dulus is smaller than 1. If the set found has cardinality one, we are \+ done. It actually turns out that this is the case for all our configur ations but the connected graphs where a `ghost' singularity has to be \+ elimanated by an external argument --see [FlaNo97]. In our case:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "rhonums:=[fsolve(Omega[0],z, complex)];\nOmega[1]:=map(proc(x) if abs(x)<1 then x fi end,rhonums); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(rhonumsG7%$!+2LGI>!\"*$\"+p5&e@ \"!#5$\"++#)p3MF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&OmegaG6#\"\" \"7#$\"+p5&e@\"!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "And a conve nient way to represent our singularity both in symbolic and numerical \+ form is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "rho_symb:=RootO f(Omega[0], z, op(Omega[1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)rh o_symbG-%'RootOfG6$,*\"\"%\"\"\"%#_ZG!#K*$F+\"\"#!\")*$F+\"\"$\"\"&$\" +p5&e@\"!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Since in our case \+ the dominant coefficient of the equation in " }{XPPEDIT 18 0 "y" "I\"y G6\"" }{TEXT -1 76 " does not vanish, the function remains finite at i ts singularity. Its value " }{XPPEDIT 18 0 "tau=Limit(y(z),z=rho)" "/% $tauG-%&LimitG6$-%\"yG6#%\"zG/F*%$rhoG" }{TEXT -1 22 " is also the qua ntity " }{XPPEDIT 18 0 "c[0]" "&%\"cG6#\"\"!" }{TEXT -1 32 " defined a bove. Since the point " }{XPPEDIT 18 0 "P(rho,tau)" "-%\"PG6$%$rhoG%$t auG" }{TEXT -1 29 " is a singularity, we have " }{XPPEDIT 18 0 "E(rh o,tau)=0" "/-%\"EG6$%$rhoG%$tauG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "diff(E(z,y),y)[rho,tau]=0" "/&-%%diffG6$-%\"EG6$%\"zG%\"yGF+6$%$ rhoG%$tauG\"\"!" }{TEXT -1 23 ". Candidate values for " }{XPPEDIT 18 0 "tau" "I$tauG6\"" }{TEXT -1 35 " are therefore obtained as follows: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "deq:=subs(z=rho_symb, \+ diff(eq,y));\nprint(`Candidate values for function at singularity`);\n tau_vals:=[fsolve(deq,y)];\nprint(tau_vals);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$deqG,**$%\"yG\"\"#\"\"$*&,(-%'RootOfG6$,*\"\"%\"\"\" %#_ZG!#K*$F2F(!\")*$F2F)\"\"&$\"+p5&e@\"!#5!\"\"*$F,F(F1!\"$F1F1F'F1F( F,F1F)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%MCandidate~values~for~fun ction~at~singularityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)tau_valsG7 $$\"+?q\")o&)!#5$\"+W(>V@\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$ \"+?q\")o&)!#5$\"+W(>V@\"!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "Plugging back these candidates into the equation eq, we could get the correct one from a carefully controlled numerical analysis:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "eTau:=subs(z=rho_symb, eq); \n[seq( evalf(subs(y=tau_vals[i], eTau)), i=1..nops(tau_vals))];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eTauG,**$%\"yG\"\"$\"\"\"*&,(-%'Roo tOfG6$,*\"\"%F)%#_ZG!#K*$F1\"\"#!\")*$F1F(\"\"&$\"+p5&e@\"!#5!\"\"*$F, F4F)!\"$F)F)F'F4F)*&,&F,F)F(F)F)F'F)F)F;F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+P!\\LG#!#6$!#pF&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "To get the constant " }{XPPEDIT 18 0 "c[1]" "&%\"cG6#\"\" \"" }{TEXT -1 72 " we are missing, let us just compute the Puiseux exp ansions verified by " }{XPPEDIT 18 0 "y(z)" "-%\"yG6#%\"zG" }{TEXT -1 56 " with the algeqtoseries procedure from the gfun package:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "all_puis:=algeqtoseries(subs (z=rho_symb*(1-t^2),eq),t,y,6);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%) all_puisG7$++%\"tG,(#\"#D\"#P\"\"\"-%'RootOfG6$,*\"\"%F,%#_ZG!#K*$F2\" \"#!\")*$F2\"\"$\"\"&$\"+p5&e@\"!#5#F,F+*$F-F5#!\"#F+\"\"!,$**,(!#DF,F -!\"\"F>F5F,F-F,,*!#7F,F-!#\\F>!\"%*$F-F8F1F,,,F-\"$`&!$K%F,F>!%B>FK!$ 5#*$F-F1\"$O\"FFFF\"\"#,$**FDF,F-F5,8F>!)cZ=iF-!)QMsL*$F-\"\"*!'oH:*$F -\"\")!(+p[\"*$F-\"\"(\"(1Vq$*$F-\"\"'\"))y%e=*$F-F9!)'o55#FK!)=\"F,F,FL!\"$F@\"\"%-%\"OG6#F,\"\"',0#\"#VF+F ,F-#\"#=F+F>#!#N\"#u*&-F.6#,*F4\"%p8F>\"%!H&!$G#F,F-\"$\")*F,F'F,F,*&, (F>#\"%l8Fip#!#)*FipF,F-#!$k'FipF,F'F5F,*&,(*&FfpF,F-F,#\"%dY\"&`1&*&F -F5FfpF,#!%q^FjqFfp#!&T[#FjqF,F'F8F,-Fjo6#*$F'#FjnF5F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "and select those with a " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 82 " term --which corresponds to the square-r oot sought due to the change of variable " }{XPPEDIT 18 0 "z=rho*(1-t^ 2)" "/%\"zG*&%$rhoG\"\"\",&\"\"\"F&*$%\"tG\"\"#!\"\"F&" }{TEXT -1 1 ": " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 252 "#--convert into series i.e. remove the O()\nall_puis_s:=map(eval, map2(subs, O=0, all_puis)) ;\n#--collect the coeff in t and select the non-null one(s)\nc1:=map(p roc(x) if x<>0 then x fi end, map(coeff, all_puis_s, t, 1));\nprint(`C onstant`, evalf(c1,10));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+all_pui s_sG7$+)%\"tG,(#\"#D\"#P\"\"\"-%'RootOfG6$,*\"\"%F,%#_ZG!#K*$F2\"\"#! \")*$F2\"\"$\"\"&$\"+p5&e@\"!#5#F,F+*$F-F5#!\"#F+\"\"!,$**,(!#DF,F-!\" \"F>F5F,F-F,,*!#7F,F-!#\\F>!\"%*$F-F8F1F,,,F-\"$`&!$K%F,F>!%B>FK!$5#*$ F-F1\"$O\"FFFF\"\"#,$**FDF,F-F5,8F>!)cZ=iF-!)QMsL*$F-\"\"*!'oH:*$F-\" \")!(+p[\"*$F-\"\"(\"(1Vq$*$F-\"\"'\"))y%e=*$F-F9!)'o55#FK!)=\"F,F,FL!\"$F@\"\"%,.#\"#VF+F,F-#\"#=F+F>#!#N\" #u*&-F.6#,*F4\"%p8F>\"%!H&!$G#F,F-\"$\")*F,F'F,F,*&,(F>#\"%l8Fep#!#)*F epF,F-#!$k'FepF,F'F5F,*&,(*&FbpF,F-F,#\"%dY\"&`1&*&F-F5FbpF,#!%q^FfqFb p#!&T[#FfqF,F'F8F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#c1G7#-%'RootO fG6#,**$%#_ZG\"\"#\"%p8*$-F'6$,*\"\"%\"\"\"F+!#KF*!\")*$F+\"\"$\"\"&$ \"+p5&e@\"!#5F,\"%!H&!$G#F3F/\"$\")*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%)ConstantG7#$!+mb=$\\\"!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 " Should we have found several expansions containing a " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 7 " term, " }{TEXT -1 58 "the right one cou ld have been selected from the formula " }{XPPEDIT 18 0 "c[1]=-sqrt( 2*rho*p[1,0]/p[0,2])" "/&%\"cG6#\"\"\",$-%%sqrtG6#**\"\"#\"\"\"%$rhoGF -&%\"pG6$\"\"\"\"\"!F-&F06$F3\"\"#!\"\"F7" }{TEXT -1 6 " and " } {XPPEDIT 18 0 "p[i,j]=Diff(eq,z$i,y$j)[rho,tau]/(factorial(i)*factoria l(j)" "/&%\"pG6$%\"iG%\"jG*&&-%%DiffG6%%#eqG-%\"$G6$%\"zGF&-F/6$%\"yGF '6$%$rhoG%$tauG\"\"\"*&-%*factorialG6#F&F8-F;6#F'F8!\"\"" }{TEXT -1 54 ". Also note that the minus sign has to be adopted for " }{XPPEDIT 18 0 "c[1]" "&%\"cG6#\"\"\"" }{TEXT -1 0 "" }{TEXT -1 59 " since the g enerating function increases with its argument." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "We can locally plot the algebraic curve in the neigh borhood of the singularity:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "implicitplot(eq, z=-.1..0.15, y=-1..1.3, numpoints=5000);" }} {PARA 13 "" 1 "" {INLPLOT "6$-%'CURVESG6e^l7$7$$!31+++++++5!#=$\"3X2lP 1Io_\"*F*7$$!3eXK9P#pFr*!#>$\"3*[k\"[;uOs\"*F*7$7$$!3hhO%RK()yk*F0$\"3 6\\]qs?>z\"*F*F-7$F47$$!3-,$p()[5oQ*F0$\"3nY%\\*H!Hk>*F*7$7$$!3WDt)yku dH*F0$\"3anmV*)F0$\"3[wV`2PmK#*F*FI7$FO7$$!3XDXe2BhN()F 0$\"3p9J?FS@X#*F*7$7$$!3aZYx&H\\:f)F0$\"3)fi$R0>ff#*F*FU7$Fen7$$!3l\\n ?4*p.T)F0$\"3S$RnEoM*p#*F*7$7$$!3)*4$=(>mVR#)F0$\"3bAZK*)*>mG*F*F[o7$F ao7$$!3=r'R$H=O&3)F0$\"3K6:;64([H*F*7$7$$!3Vs>mVRK()yF0$\"3B2\")\\o;s8 $*F*Fgo7$F]p7$$!3^L1'*HSegxF0$\"3$)GYq(**=+K*F*7$7$$!3[Lcgn7@NvF0$\"3. ;p=1d'3M*F*Fcp7$Fip7$$!3EzN#)35.OuF0$\"3')p.B!)QPX$*F*7$7$$!3#fH\\:f)4 $=(F0$\"3i:Y:I\\,o$*F*F_q7$Feq7$$!3M%zt([ap6rF0$\"3R)Rcp#)G4P*F*7$7$$! 3OeH\\:f)4$oF0$\"3^'*oh.]7&R*F*F[r7$Far7$$!38N#*enxc(y'F0$\"3@L%QA-vmR *F*7$7$$!3T>mVRK()ykF0$\"3h2%on)G9A%*F*Fgr7$F]s7$$!3))*HeS`NOY'F0$\"3b /=OH5gA%*F*7$7$$!3fG9RBux%G'F0$\"3\"*f)4$=(>mV*F*Fcs7$Fis7$$!39G[1Z1eR hF0$\"3WT'*H&>9%[%*F*7$7$$!3&=G!Qj0wEhF0$\"3K.%p&)R?-X*F*F_t7$Fet7$$!3 \"\\\\S7(QY:eF0$\"3D&38NUqTZ*F*7$7$$!3IWRK()ykudF0$\"3IrfRJp&)z%*F*F[u 7$7$$!3gVRK()ykudF0$\"3>qfRJp&)z%*F*7$$!3G/$yXK!p\"\\&F0$\"3]-dbACC+&* F*7$7$$!3N0wE6_`AaF0$\"3C\"3EkxY(4&*F*F\\v7$7$$!3l/wE6_`AaF0Fev7$$!3B] @8!fs#o^F0$\"3#>2P)oq&* F*Fgw7$7$$!3BI\\:f)4$=ZF0F`x7$$!3f4%Q;)ocAXF0$\"3_\"\\m4%)z/e*F*7$7$$! 3G\"f)4$=(>mVF0$\"33t;#y%on+'*F*Ffx7$7$F]y$\"3>u;#y%on+'*F*7$$!38*=$e( *)=.?%F0$\"3^?vgd`&zg*F*7$7$$!3s`A/2X39SF0$\"32fo)R$)o7j*F*Fey7$F[z7$$ !3'H6]nx4&yQF0$\"3Q?HTGY$ej*F*7$7$$!3Z:f)4$=(>m$F0$\"3CPA5)Q\"*=m*F*Fa z7$Fgz7$$!3m&*y\"*Qo:k'*F*7$7$$!3Ax&H\\:f)4LF0$\"3quw $G[PCp*F*F][l7$Fc[l7$$!3sm/`#QrjB$F0$\"3Q3XnHY&Hp*F*7$7$$!3mRK()ykudHF 0$\"3>^!yNp`Fs*F*Fi[l7$F_\\l7$$!3u^.]&>nh\"HF0$\"3F0$\"3C%=e4D:`!)*F*7$7$$!3DDUq]%39!>F0$\"3Ihw# )H*Rx\")*F*F\\_l7$Fb_l7$$!3Tg!\\h0odi\"F0$\"3RpFEzy\"4$)*F*7$7$$!3M()y kudH\\:F0$\"3*y!)GiTU:&)*F*Fh_l7$F^`l7$$!3Q)yy'fEu+8F0$\"3E]>[w\"Qe&)* F*7$7$$!3E\\:f)4$=(>\"F0$\"3f\\)=e$p+'))*F*Fd`l7$Fj`l7$$!3MkVy*p!GQ(*! #?$\"37e%HfT?!z)*F*7$7$$!3$=6_`A/2X)Fcal$\"3FW$z`7y8#**F*F`al7$Fgal7$$ !3p?9;L:N8kFcal$\"3Cx>d%fqq*)*F*7$7$$!3xK()ykudH\\Fcal$\"3b:r\"p&f^e** F*F]bl7$7$Fdbl$\"3m;r\"p&f^e**F*7$$!3+c>4yzT9GFcal$\"39_ `A/2X39FcalFecl7$7$Fjcl$\"3'=Fcal$\"3>xHx#o))=/\"Fgcl7$7$Fjcl$\"3b\"R(> px.V5FgclFefl7$7$$\"3v39!p\"G!Qj&Fcal$\"3wrD/f0Wj`J Fcal$\"3!)e)4$=(>mV*F*7$7$$\"3+\"=&>0Wj`JFcal$\"3pd)4$=(>mV*F*Ficl7$7$ Fjcl$\"33sTm*y`0+\"Fgcl7$$\"3>%4I.)R!RR#Fcal$\"34DYK2M'e+\"Fgcl7$7$F`g l$\"3&H))\\CT\\a+\"FgclFchl7$F[gl7$$\"3(G#)=00njK%Fcal$\"3$[Dt%zM(G0\" Fgcl7$7$$\"34())f`997W&Fcal$\"3@=(>mVRK2\"FgclF]il7$Fcil7$$\"3[4>/*e2T e&Fcal$\"3cDgrKmpt5Fgcl7$7$F`gl$\"3@RFA$=)*R2\"FgclFiil7$7$$\"3o)ykudH \\:*Fcal$\"3ST*Q0\"yBq!*F*7$$\"3qBdvBSsgzFcal$\"3CG!Qj0wE6*F*7$FhjlF_g l7$Fihl7$$\"3MQDwr'pU\"eFcal$\"3nH1;4/z15Fgcl7$7$$\"3i!ySZA+_&zFcalF`e lF_[m7$Fe[m7$$\"3$z&3Tl')yk*)Fcal$\"3:[ih++?55Fgcl7$7$Fdjl$\"3&e*z-GY^ 45FgclFi[m7$F_jl7$$\"3B5)f$))*3&G\")Fcal$\"3-%[Rg]#o#3\"Fgcl7$7$$\"3T! zkudH\\:*Fcal$\"3-$H]<#e5!4\"FgclFc\\m7$Fcjl7$$\"3Q4*ouD2*45F0$\"3)HR$ f+\\\"e-*F*7$7$$\"3&p\"G!Qj0wE\"F0$\"30Yh*>dkv\"*)F*F_]m7$F_\\m7$$\"3c $=c9%zt)>\"F0$\"3NU7#>`'y95Fgcl7$7$Ff]m$\"3p?JL:Lf85FgclF[^m7$7$FdjlF \\]m7$$\"3)H,jWqxQ.\"F0$\"3L!)*=hSUZ4\"Fgcl7$7$$\"3)>%H2)*3r]6F0$\"3\\ ,p\"G!Qj06FgclFf^m7$F\\_m7$$\"3.9cY9#=7C\"F0$\"36CfRX9136Fgcl7$7$Ff]m$ \"3n/=da!=#46FgclFb_m7$7$$\"3&[:f)4$=(>;F0$\"3pi*eoVUev)F*7$$\"3QOw;?/ R.:F0$\"3e'>mVRK()y)F*7$7$$\"3cOw;?/R.:F0$\"3p(>mVRK()y)F*Fe]m7$Fa^m7$ $\"3sG=5Fgcl7$7$F]`m$\"3B5[Q&f^u,\"FgclF]am7$7$ $\"37D@5FgclFccm7$F`bm7$$ \"3#fc)f%o-\"fx9T2LO6FgclF]dm7$F]c m7$$\"3!4R4p/!pMAF0$\"3#=\"=De?!pa)F*7$7$$\"3,J=(>mVRK#F0$\"3\")eq^B#Q k^)F*Fgdm7$Ficm7$$\"3fjBD`u;Z@F0$\"3)HaFa69Z-\"Fgcl7$7$F^em$\"3&\\(opl )\\]-\"FgclFcem7$7$F^cm$\"3[>x9T2LO6Fgcl7$$\"3f`$)oN\")p&)>F0$\"3H\") \\B\"R_n8\"Fgcl7$7$$\"3Znf.]P?2?F0$\"3b%39!p\"G!Q6FgclF`fm7$Fffm7$$\"3 8wXr)HeME#F0$\"3$>vLC#GfV6Fgcl7$7$F^em$\"3%Ga623'fX6FgclF\\gm7$7$$\"3# *o\"G!Qj0wEF0$\"3Uq-9Zss6%)F*7$$\"3s_T;GYS^CF0$\"3-mVRK()yk%)F*7$F[hm7 $F^em$\"3#*fq^B#Qk^)F*7$7$F^em$\"3tuopl)\\]-\"Fgcl7$$\"3rc(Qc13]Y#F0$ \"3(z<&GXr'y-\"Fgcl7$7$Fggm$\"3Q!\\5](*p)G5FgclFhhm7$Fbgm7$$\"3r8bvux? 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The values returned are: \n1.the sym bolic expressions for the " }{XPPEDIT 18 0 "c[1] " "&%\"cG6#\"\"\"" } {TEXT -1 37 " constant\n2.the dominant singularity " }{XPPEDIT 18 0 "r ho" "I$rhoG6\"" }{TEXT -1 0 "" }{TEXT -1 24 "\n3.the Puiseux expansion " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1012 "singExpansion:=proc(e q, y, z)\nlocal rho_symb, \n deq, tau_vals, \n all_pui s, all_puis_s, c1;\n\n#--numerical and symbolic representation of the \+ singularities\nrho_symb:=locateDominantSing(eq, y, z);\n\n#--values of the function at the singularity\ndeq:=subs(z=rho_symb, diff(eq,y));\n print(`Candidate values for function at singularity`);\ntau_vals:=[fso lve(deq,y)];\nprint(tau_vals);\n\n#--Puiseux expansion; we expect a si ngle exp. to have a t term\nall_puis:=algeqtoseries(subs(z=rho_symb*(1 -t^2),eq),t,y,6);\n\n#--convert into series i.e. remove the O()\nall_p uis_s:=map(eval, map2(subs, O=0, all_puis));\n#--collect the coeff in \+ t and select the non-null one(s)\nc1:=map(proc(x) if x<>0 then x fi en d, map(coeff, all_puis_s, t, 1));\n\nif nops(c1)<>1 then\n print(`Prob lem in singular expansion`); print(all_puis); RETURN(FAIL)\nelse\n c1: =op(c1); if evalf(c1)>0 then c1:=-c1 fi\nfi;\n\n#--returns the constan t c1 and the singularity in symbolic forms\nprint(`c1 constant`, c1,ev alf(c1,20));\n[c1,rho_symb, my_puis]\nend:" }}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 43 "Applications to non-crossing configurations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 170 "We now present the whole chain that goes from the grammars specification to the asymptotic machinery. More pre cisely, for each of the NC configurations studied above, we:" }}{PARA 0 "" 0 "" {TEXT -1 127 "1.call Combstruct[gfeqns] to retrieve the syst em of equations verified by the generating functions associated with t he grammar," }}{PARA 0 "" 0 "" {TEXT -1 98 "2.compute the algebraic eq uation verified by a given generating function with gfun[algfuntoalgeq ]," }}{PARA 0 "" 0 "" {TEXT -1 66 "3.feed this equation to the asympto tic machinery developped above." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 5 "Trees" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Generating functions a ssociated with the grammar:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "gfeqns(tbf, unlabelled, z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&/ -%\"BG6#%\"zG*&-%\"FGF'\"\"#-%\"ZGF'\"\"\"/F**$,&F/F/F%!\"\"F3/F-F(/-% \"TGF'*&F-F/F2F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "treesSy s:=gfsolve(tbf, unlabelled, z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%) treesSysG<&/-%\"FG6#%\"zG-%'RootOfG6#,(%#_ZG!\"\"*&F/\"\"$F*\"\"\"F3F3 F3/-%\"TGF),$*&F*F3,&F0F3*&F+\"\"#F*F3F3F0F0/-%\"BGF)F:/-%\"ZGF)F*" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Algebraic equation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "trees:=algfuntoalgeq(subs(treesSys, T(z)), y(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&treesG,(*&%\"yG\" \"\"%\"zGF(!\"\"*$F)\"\"#F(*$F'\"\"$F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Asymptotic machinery:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "resTrees:=singExpansion(trees, y, z):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/Singularity~isG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"%\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%MCandidate~values~fo r~function~at~singularityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$!+AAA AA!#5$\"+AAAAAF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%,c1~constantG-%'R ootOfG6#,&!\"%\"\"\"*$%#_ZG\"\"#\"$V#$!5cOo\"*>)f+IG\"!#?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Plot of the singularity:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "implicitplot(trees, z=-0.2..0.2, y= -1..1, numpoints=5000);" }}{PARA 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"6#-%'RootOfG6 $,*\"\"%\"\"\"%#_ZG!#K*$F)\"\"#!\")*$F)\"\"$\"\"&$\"+p5&e@\"!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%MCandidate~values~for~function~at~sin gularityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+?q\")o&)!#5$\"+W(>V @\"!\"*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6%%,c1~constantG-%'RootOfG6#, **$%#_ZG\"\"#\"%p8*$-F%6$,*\"\"%\"\"\"F)!#KF(!\")*$F)\"\"$\"\"&$\"+p5& e@\"!#5F*\"%!H&!$G#F1F-\"$\")*$!5(p/4Q%ob=$\\\"!#?" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 16 "Connected graphs" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Generating functions associated with the grammar:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "CGSys:=gfsolve(ar, unlabelle d, z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&CGSysG<%/-%\"ZG6#%\"zGF*/ -%#EAGF)-%'RootOfG6#,*%#_ZG\"\"\"*$F2\"\"#!\"$*$F2\"\"$F5F*!\"\"/-%\"C GF),$*(F*F3,(F3F3F.F6*$F.F5F5F3,*F9F3F.F8F@!\"#F*F3F9F9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Algebraic equation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "CG:=algfuntoalgeq(subs(CGSys,C(z)), y(z));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#CGG,**$%\"yG\"\"$\"\"\"*$F'\"\"#F) *&F'F)%\"zGF)!\"$*$F-F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Asym ptotic machinery:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "resCG: =singExpansion(CG, y, z):" }}{PARA 8 "" 1 "" {TEXT -1 82 "Error, (in l ocateDominantSing) More than one root of modulus < 1: not implemented! 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C6Fdjp-%'COLOURG6&%$RGBG\"\"\"\"\"!F[_w-%+AXESLABELSG6$%\"zG%\"yG" 2 256 252 252 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 3 0 0 0 0 0 0 }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 11 "Dissections" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "dissSys:=gfsolve(dissG, unlabelled, z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(dissSysG<$/-%#DiG6#%\"zG,(#\"\"\"\"\"%F-F*F,*$,(F-F- F*!\"'*$F*\"\"#F-#F-F3#!\"\"F./-%\"ZGF)F*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 47 "diss:=algfuntoalgeq(subs(dissSys,Di(z)), y(z));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dissG,(*$%\"yG\"\"#F(*&,&!\"\"\"\" \"%\"zGF+F,F'F,F,F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "re sDiss:=singExpansion(diss, y, z):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% /Singularity~isG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RootOfG6$,(\"\" \"F'%#_ZG!\"'*$F(\"\"#F'$\"+`(Gdr\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%MCandidate~values~for~function~at~singularityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#$\"+)=K*GH!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 %%,c1~constantG-%'RootOfG6#,(*$%#_ZG\"\"#\"\")-F%6$,(\"\"\"F/F)!\"'F(F /$\"+`(Gdr\"!#5\"\"$!\"\"F/$!5n4SN_x&GHY#!#?" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 118 "It should be noticed that the equation obtained in the paper and stated below in slightly different.But remember that " } {XPPEDIT 18 0 "z^i" ")%\"zG%\"iG" }{TEXT -1 57 " counts here the numbe r of dissections of a polygon with " }{XPPEDIT 18 0 "i+1" ",&%\"iG\"\" \"\"\"\"F$" }{TEXT -1 57 " vertices. 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But only the first one makes sense since the generating function increases with its argument. 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It is interesting to obs erve that these values are within about 2% in any case:" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 5 "Trees" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "exactTr:=evalf(count([T,tbf], size=100));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(exactTrG$\"5zCJx*[UH76$\"#f" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "estimTr:=nbEnt(resTrees[1],resTrees [2],100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(estimTrG$\"5xtK\\\\TB= $3$\"#f" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "exactTr/estimTr; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5uX()p%oz'445!#>" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "Forests" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "exactFo:=evalf(count([F,fo], size=100));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(exactFoG$\"5gkKDp;fP%Q\"\"#o" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "estimFo:=nbEnt(resForests[1], resFo rests[2], 100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(estimFoG$\"5RMoR + " 0 "" {MPLTEXT 1 0 16 "exactFo/estim Fo;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5M7\\#)=iZ265!#>" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 16 "Connected graphs" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "exactCo:=evalf(count([C,ar],size=100));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(exactCoG$\"5I&QK+3d$****[\"#x" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "estimCo:=nbEnt(resCG[1],resC G[2],100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(estimCoG$\"5Wl*>&)fjX V#[\"#x" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "exactCo/estimCo; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5WOGV.b/o:5!#>" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 14 "General graphs" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "exactGG:=evalf(count([G, br], size=100));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(exactGGG$\"5(RU6'ew#)\\Fb\"##)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "estimGG:=nbEnt(resGG[1], res GG[2], 100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(estimGGG$\"5!*)R7)* Q#[SDa\"##)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "exactGG/esti mGG;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5\"e_k4am<)=5!#>" }}}} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 11 "Dissections" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 47 "exactDiss:=evalf(count([Di, dissG], size=100)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*exactDissG$\"57?I#ocbFL]#\"#` " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "estimDiss:=nbEnt(resDis s[1], resDiss[2], 100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*estimDis sG$\"5O4&QVrb-X\\#\"#`" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "e xactDiss/estimDiss;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5;i&)[!*yx`. 5!#>" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 11 "Partitions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "exactPart:=evalf(count([P, partG], \+ size=100));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*exactPartG$\"5p'\\J, 4Z*>l*)\"#P" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "estimPart:=n bEnt(resParts[1], resParts[2], 100);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%*estimPartG$\"5-%oDv(fq " 0 "" {MPLTEXT 1 0 20 "exactPart/estimPart;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5iYuV\"[k@')))*!#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "References" }}}{PARA 0 "" 0 "" {TEXT -1 98 "[Dr97] M. Drmota, S ystems of functional equations, Random Structures and Algorithms 10, 1 -2, 1997." }}{PARA 0 "" 0 "" {TEXT -1 131 "[FlaNo97] P. Flajolet and M . Noy, Analytic Combinatorics of Non-Crossing Configurations, Rapport \+ de Recherche INRIA No. 3196, 1997." }}{PARA 0 "" 0 "" {TEXT -1 80 "[Ha Pa73] F. Harary and E.M. Palmer, Graphical Enumeration, Academic Press , 1973." }}{PARA 0 "" 0 "" {TEXT -1 100 "[Sloa95] N.J.A. Sloane and S. Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995. " }}}{MARK "6 1 1 7 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 } A 0 "> " 0 "" {MPLTEXT 1 0 48 "estimPart:=n bEnt(resParts[1], resParts[2], 100);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%*estimPartG$\"5-%oDv(fq " 0 "" {MPLTEXT 1 0 20 "exactPart/estimPart;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5iYuV\"[k@')))*!#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "References" }}}{PARA 0 "" 0 "" {TEXT -1 98 "[Dr97] M. Drmota, S ystems of functional equations, Random Structures and Algorithms 10, 1 -2, 1997." chemist!kchemistr+ chessboardchiWchicago!kchief}childxchoic3cychomskjchoosNchordxchosen.cW chudnovskchyzak+#' y}׵ciWcircl xE circulantcircular circumstanc clarendonclas/<+ .myԡxclass +!kx classical/]FsWjclassifclassifi classificat cleanclear#y}E cloakroomclon9DclosWG<)]fW׵xElightlike[* .!>9DT!k}]xlikewis!klimitA]ylimitatElin#'linalgline'A!k7pylinear #'+KrV]cfry yW׵]linearit linearsollink 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1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 4 269 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 5 271 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 272 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 273 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 274 1 {CSTYLE "" -1 -1 " " 1 16 0 0 0 0 0 2 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 275 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 276 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 277 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 278 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 256 45 "ROBUSTNESS IN RANDOM INTERCONNECTIONS GRAPHS " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 257 36 "Philip pe Flajolet, January 11, 1998 " }}{PARA 258 "" 0 "" {TEXT -1 48 "(Base d on joint work with Kostas Hatzis, Patras)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 13 "Random graphs" }{TEXT -1 258 " are usually studied by the probabil istic method that is based on inequalities and on approximations of ra ndom variables. Many models that appear in this context can however be subjected to analytic methods. This worksheet explores one such situa tion using " }{HYPERLNK 17 "Combstruct" 2 "combstruct" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Gfun" 2 "gfun" "" }{TEXT -1 119 ", and the Maple system. The objective is to characterise the interplay between three \+ parameters of a random graph: its " }{TEXT 269 7 "density" }{TEXT -1 24 " (number of edges), its " }{TEXT 270 10 "robustness" }{TEXT -1 27 " to link failures, and its " }{TEXT 271 12 "connectivity" }{TEXT -1 16 " by short paths." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 9 "A triple " }{XPPEDIT 18 0 "``(Gamma,s,t)" "-%!G6%%&Gamma G%\"sG%\"tG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "Gamma" "I&GammaG6 \"" }{TEXT -1 14 " is a graph, " }{XPPEDIT 18 0 "s" "I\"sG6\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 69 " are tw o designated nodes (the source and the target), is said to be " } {XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 11 "-robust if " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "t" "I\"tG6\"" } {TEXT -1 72 " are connected by at least two edge-disjoint vertices of \+ length exactly " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 85 ". This de finition captures an intuitive notion of robustness to link failures, \+ since " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 148 " will remain connected by \"short\" paths even in the event of a link (i.e., edge) becoming unavailable \+ in the interconnection graph represented by " }{XPPEDIT 18 0 "Gamma" "I&GammaG6\"" }{TEXT -1 3 ". \n" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 265 19 "random graph model " }{TEXT -1 11 "is that of " } {XPPEDIT 18 0 "G[n,p]" "&%\"GG6$%\"nG%\"pG" }{TEXT -1 21 " where the g raph has " }{XPPEDIT 18 0 "n " "I\"nG6\"" }{TEXT -1 36 " vertices and \+ each of the possible " }{XPPEDIT 18 0 "n*(n-1)/2" "*(%\"nG\"\"\",&F#F $\"\"\"!\"\"F$\"\"#F'" }{TEXT -1 47 " edges is independently taken wit h probability " }{XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 178 ". Under t his model, and because of the symmetry it implies with respect to the \+ naming of nodes, it is strictly equivalent to examine robustness prope rties between a fixed source " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 17 " and destination " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 35 " or between a random pair of nodes " }{XPPEDIT 18 0 "s,t" "6$%\"sG%\"t G" }{TEXT -1 77 ". For definiteness, we adopt the latter formulation. \+ A graph with very samll " }{XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 40 " will not be robust, while a graph with " }{XPPEDIT 18 0 "p" "I\"pG6\" " }{TEXT -1 123 " close to 1 will have a high number of edge-disjoint \+ paths of small length. The purpose is to evaluate the threshold value \+ " }{XPPEDIT 18 0 "p[0]" "&%\"pG6#\"\"!" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 13 " from which " }{XPPEDIT 18 0 "l" "I \"lG6\"" }{TEXT -1 134 "-robustness becomes likely. In this worksheet, we focus on the problem of estimating the mean number of edge-disjoin t pairs of length " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 82 " betwe en a random source and a random target, and deduce the associated thre shold." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The analysis proceeds in stages:" }}{PARA 0 "" 0 "" {TEXT -1 6 " - \+ " }{TEXT 264 6 "Step 1" }{TEXT -1 51 ". Enumerate the number of pairs of paths of length " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 18 " con necting nodes " }{XPPEDIT 18 0 "1" "\"\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "l+1" ",&%\"lG\"\"\"\"\"\"F$" }{TEXT -1 57 " on a line, \+ so that both paths use the same set of nodes " }{XPPEDIT 18 0 "1,2,`.. .`,l+1" "6&\"\"\"\"\"#%$...G,&%\"lG\"\"\"\"\"\"F(" }{TEXT -1 47 ".This is really a problem of counting so-called" }{TEXT 296 23 " avoiding p ermutations " }{TEXT -1 147 "defined as cyclic permutations with const raints on their so-called \"succession gaps\", i.e., the values of the differences between succesive values " }{XPPEDIT 18 0 "sigma[i+1]-sig ma[i]" ",&&%&sigmaG6#,&%\"iG\"\"\"\"\"\"F(F(&F$6#F'!\"\"" }{TEXT -1 48 " that are constrained not to belong to the set " }{XPPEDIT 18 0 " \{-1,0,+1\}" "<%,$\"\"\"!\"\"\"\"!\"\"\"" }{TEXT -1 36 ". This steps i tself decomposes into:" }}{PARA 0 "" 0 "" {TEXT -1 11 " S" } {TEXT 267 6 "tep 1a" }{TEXT -1 22 ". Enumerate so-called " }{TEXT 273 9 "templates" }{TEXT -1 44 " that are skelettons of permutations with \+ a " }{TEXT 275 31 "distinguished set of exceptions" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 10 " " }{TEXT 268 7 "Step 1b" } {TEXT -1 140 ". Transform the counting of templates into enumeration o f permutations with a distinguished set of exceptions and conclude by means of the " }{TEXT 274 32 "Principle of Inclusion-Exclusion" } {TEXT -1 38 ", a classical combinatorial principle." }}{PARA 0 "" 0 " " {TEXT -1 6 " - " }{TEXT 266 6 "Step 2" }{TEXT -1 78 ". Modify the model in order to allow for nodes taken from outside the segment " } {XPPEDIT 18 0 "1,2,`...`,l+1" "6&\"\"\"\"\"#%$...G,&%\"lG\"\"\"\"\"\"F (" }{TEXT -1 16 ". We then count " }{TEXT 308 14 "avoiding paths" } {TEXT -1 175 " that may borrow \"outer nodes\". This situation models \+ the original random graph problem by allowing nodes taken from the who le pool of nodes that are available from the graph " }{XPPEDIT 18 0 "G amma" "I&GammaG6\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 7 " \+ - " }{TEXT 276 6 "Step 3" }{TEXT -1 153 ". Return to the original ra ndom graph problem and obtain the expected number of edge-disjoint pai rs between a source and a destination in a random graph " }{XPPEDIT 18 0 "Gamma" "I&GammaG6\"" }{TEXT -1 16 " that obeys the " }{XPPEDIT 18 0 "G[n,p]" "&%\"GG6$%\"nG%\"pG" }{TEXT -1 7 " model." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 10 "References" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 " [Comtet, 1974] : L. Comtet, " }{TEXT 259 22 "Advanced Combina torics" }{TEXT -1 15 ", Reidel, 1974." }}{PARA 0 "" 0 "" {TEXT -1 38 " [EIS]: N. Sloane and S. Plouffe, " }{TEXT 260 37 "The Encyclopedi a of Integer Sequences" }{TEXT -1 23 ", Academic Press, 1995." }} {PARA 0 "" 0 "" {TEXT -1 81 " This Maple worksheet is based on the current versions of the Maple packages " }{TEXT 262 11 "combstruct " }{TEXT -1 4 "and " }{TEXT 263 5 "gfun " }{TEXT -1 42 "(for version V.4 ) that can be found under " }{TEXT 261 30 "http://www-rocq.inria.fr/al go/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "with(combstruct);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# 71%+allstructsG%&countG%%drawG%)finishedG%'gfeqnsG%)gfseriesG%(gfsolve G%,iterstructsG%%markG%'momentG%+nextstructG%,prog_gfeqnsG%.prog_gfser iesG%-prog_gfsolveG%)varianceG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "with(gfun);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7V%(LaplaceG%.a lgebraicsubsG%.algeqtodiffeqG%.algeqtoseriesG%.algfuntoalgeqG%&borelG% .cauchyproductG%.diffeq*diffeqG%.diffeq+diffeqG%2diffeqtohomdiffeqG%,d iffeqtorecG%)guesseqnG%(guessgfG%0hadamardproductG%0holexprtodiffeqG%) invborelG%,listtoalgeqG%-listtodiffeqG%0listtohypergeomG%+listtolistG% .listtoratpolyG%*listtorecG%-listtoseriesG%5listtoseries/LaplaceG%1lis ttoseries/egfG%4listtoseries/lgdegfG%4listtoseries/lgdogfG%1listtoseri es/ogfG%4listtoseries/revegfG%4listtoseries/revogfG%,maxdegcoeffG%*max degeqnG%,maxordereqnG%,mindegcoeffG%*mindegeqnG%,minordereqnG%*options gfG%,poltodiffeqG%)poltorecG%/ratpolytocoeffG%(rec*recG%(rec+recG%,rec todiffeqG%,rectohomrecG%*rectoprocG%.seriestoalgeqG%/seriestodiffeqG%2 seriestohypergeomG%-seriestolistG%0seriestoratpolyG%,seriestorecG%/ser iestoseriesG" }}}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 48 "1. Permutations w ith constrained succession gaps" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 57 "The goal of this section is to enumerate \+ permutations of " }{XPPEDIT 18 0 "[1,`..`,n]" "7%\"\"\"%#..G%\"nG" } {TEXT -1 30 " that are cyclic, of the form " }{XPPEDIT 18 0 "[1,tau[2] ,`...`,tau[n-1],n]" "7'\"\"\"&%$tauG6#\"\"#%$...G&F%6#,&%\"nG\"\"\"\" \"\"!\"\"F," }{TEXT -1 29 ", and with no succession gap " }{XPPEDIT 18 0 "tau[j+1]-tau[j]" ",&&%$tauG6#,&%\"jG\"\"\"\"\"\"F(F(&F$6#F'!\"\" " }{TEXT -1 5 " in " }{XPPEDIT 18 0 "\{-1,0,+1\}" "<%,$\"\"\"!\"\"\" \"!\"\"\"" }{TEXT -1 28 ". We call such permutations " }{TEXT 290 21 " avoiding permutations" }{TEXT -1 80 ". In another language, what is th e number of paths of length n-1 that join 1 to " }{XPPEDIT 18 0 "n" "I \"nG6\"" }{TEXT -1 81 " and have no edge in common with those of the l ine graph defined by the interval " }{XPPEDIT 18 0 "[1,`..`,n]" "7%\" \"\"%#..G%\"nG" }{TEXT -1 2 "? " }}{PARA 0 "" 0 "" {TEXT -1 28 "There \+ are no such paths for " }{XPPEDIT 18 0 "n=2,3,4,5" "6&/%\"nG\"\"#\"\"$ \"\"%\"\"&" }{TEXT -1 68 ". Surprisingly enough, the first nontrivial \+ configurations occur at " }{XPPEDIT 18 0 "n=6" "/%\"nG\"\"'" }{TEXT -1 18 " (try it!), namely" }}{PARA 0 "" 0 "" {TEXT -1 47 " \{[1, 4 , 2, 5, 3, 6], [1, 3, 5, 2, 4, 6]\}" }}{PARA 0 "" 0 "" {TEXT -1 8 "a nd for " }{XPPEDIT 18 0 "n=7" "/%\"nG\"\"(" }{TEXT -1 9 ", one has" }} {PARA 0 "" 0 "" {TEXT -1 279 " \{[1, 3, 6, 4, 2, 5, 7], [1, 3, 5 , 2, 6, 4, 7], [1, 4, 6, 2, 5, 3, 7], [1, 4, 2, 6, 3, 5, 7], [1, 5 , 3, 6, 4, 2, 7],\n [1, 5, 3, 6, 2, 4, 7], [1, 5, 2, 4, 6, 3, \+ 7], [1, 4, 6, 3, 5, 2, 7], [1, 6, 3, 5, 2, 4, 7], [1, 6, 4, 2, 5, \+ 3, 7]\}\nso that the numbers are " }{XPPEDIT 18 0 "Q[6]=2, Q[7]=10" "6 $/&%\"QG6#\"\"'\"\"#/&F%6#\"\"(\"#5" }{TEXT -1 88 ". A brute force enu meration routine based on the predefined structures availaible under \+ " }{TEXT 289 11 "Combstruct " }{TEXT -1 15 "is given below." }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 15 " 1.1. Templates" }}{PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 277 8 "template" }{TEXT -1 68 " is a scheme tha t specifies which \"bits and pieces\" of the interval " }{XPPEDIT 18 0 "1,2,`...`,l+1" "6&\"\"\"\"\"#%$...G,&%\"lG\"\"\"\"\"\"F(" }{TEXT -1 58 " may be exceptions. We firtst need to enumerate templates." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 278 70 "In combs truct, we specify templates as decompositions of the interval " } {XPPEDIT 18 0 "1..n" ";\"\"\"%\"nG" }{TEXT 311 6 " into " }{TEXT 279 6 "blocks" }{TEXT 280 16 " that are either" }}{PARA 0 "" 0 "" {TEXT -1 24 " -- isolated points " }{XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 93 " -- blocks of contiguous u nit intervals (based at integer points) oriented left-to-right " } {XPPEDIT 18 0 "LR" "I#LRG6\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 92 " -- blocks of contiguous unit intervals (based at integer po ints) oriented right-to-left " }{XPPEDIT 18 0 "RL" "I#RLG6\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 30 "A template must start with an \+ " }{XPPEDIT 18 0 "LR" "I#LRG6\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "p " "I\"pG6\"" }{TEXT -1 33 " block and end similarly with an " } {XPPEDIT 18 0 "LR" "I#LRG6\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "p" "I \"pG6\"" }{TEXT -1 7 " block." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEX T -1 18 "For instance, for " }{XPPEDIT 18 0 "n=13 " "/%\"nG\"#8" }{TEXT -1 15 ", the template " }}{PARA 260 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[[1,2,3],[4],[5,6],[7],[8],[11,10,9],[1 2,13]" "7)7%\"\"\"\"\"#\"\"$7#\"\"%7$\"\"&\"\"'7#\"\"(7#\"\")7%\"#6\"# 5\"\"*7$\"#7\"#8" }}{PARA 0 "" 0 "" {TEXT -1 97 "will correspond to an y cyclic permutation that has successions of values (in the cycle trav ersal)" }}{PARA 261 "" 0 "" {TEXT -1 49 " 1,2; 2,3; 5,6; 11,1 0; 10,9; 12,13" }}{PARA 0 "" 0 "" {TEXT -1 77 "as distinguished \+ exceptions to the basic constraint of avoiding permutations." }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 31 "The combinatorial specification" }} {PARA 262 "" 0 "" {TEXT -1 39 "First a preliminary specification. Let \+ " }{XPPEDIT 18 0 "\{a,b\}" "<$%\"aG%\"bG" }{TEXT -1 84 " be a binary a lphabet. The collection of strings beginning and ending with a letter \+ " }{XPPEDIT 18 0 "a" "I\"aG6\"" }{TEXT -1 16 " is described by" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "sp0:=S=Prod!(Sequence(Prod(a, Sequence(b))),a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sp0G/%\"SG-%%P rodG6$-%)SequenceG6#-F(6$%\"aG-F+6#%\"bGF/" }}}{PARA 0 "" 0 "" {TEXT -1 69 "(It suffices to decompose according to each occurrence of the l etter " }{XPPEDIT 18 0 "a" "I\"aG6\"" }{TEXT -1 2 ".)" }}{PARA 0 "" 0 "" {TEXT -1 61 "Now, the three types of blocks in a template are descr ibed by" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "sp1:=Prod(begin_ blockP,Z,end_blockP); sp2:=Prod(begin_blockLR,Z,Sequence(Prod(mu_lengt h,Z),card>=1),end_blockLR);\nsp3:=Prod(begin_blockRL,Sequence(Prod(mu_ length,Z),card>=1),Z,end_blockRL);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%$sp1G-%%ProdG6%%-begin_blockPG%\"ZG%+end_blockPG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sp2G-%%ProdG6&%.begin_blockLRG%\"ZG-%)SequenceG6$ -F&6$%*mu_lengthGF)1\"\"\"%%cardG%,end_blockLRG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sp3G-%%ProdG6&%.begin_blockRLG-%)SequenceG6$-F&6$%*m u_lengthG%\"ZG1\"\"\"%%cardGF/%,end_blockRLG" }}}{PARA 0 "" 0 "" {TEXT -1 9 "Clea"rly, " }{XPPEDIT 18 0 "sp2" "I$sp2G6\"" }{TEXT -1 5 " \+ and " }{XPPEDIT 18 0 "sp3" "I$sp3G6\"" }{TEXT -1 148 " are combinatori ally isomorphic. For reasons related to application of the inclusion e xclusion argument, we keep track of the size (number of nodes " } {XPPEDIT 18 0 "l=1" "/%\"lG\"\"\"" }{TEXT -1 78 " of the basic interva l graph) as well as of the length of blocks and of their " }{XPPEDIT 18 0 "LR" "I#LRG6\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "RL" "I#RLG6\" " }{TEXT -1 46 " character. Then, the grammar of templates is:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 187 "Q:=subs([a=Union(sp1,sp2),b =sp3],sp0), begin_blockP=Epsilon, end_blockP=Epsilon, begin_blockLR=Ep silon, end_blockLR=Epsilon, begin_blockRL=Epsilon, end_blockRL=Epsilon , mu_length=Epsilon;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"QG6*/%\"SG -%%ProdG6$-%)SequenceG6#-F)6$-%&UnionG6$-F)6%%-begin_blockPG%\"ZG%+end _blockPG-F)6&%.begin_blockLRGF6-F,6$-F)6$%*mu_lengthGF61\"\"\"%%cardG% ,end_blockLRG-F,6#-F)6&%.begin_blockRLGF;F6%,end_bl#ockRLGF0/F5%(Epsilo nG/F7FK/F:FK/FCFK/FHFK/FIFK/F?FK" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "temp15:=draw([S,\{Q\},unlabelled],size=15);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'temp15G-%%ProdG6$-%)SequenceG6)-F&6$-F&6% %-begin_blockPG%\"ZG%+end_blockPG%(EpsilonGF+-F&6$-F&6&%.begin_blockLR GF0-F)6%-F&6$%*mu_lengthGF0F:F:%,end_blockLRGF2F+F+-F&6$F--F)6#-F&6&%. begin_blockRLG-F)6$F:F:F0%,end_blockRLG-F&6$-F&6&F7F0-F)6#F:F=F2F-" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "decode:=proc(e) subs([Prod =proc() args end,Sequence=proc() args end,mu_length=NULL,Epsilon=NULL] , e); [\"] end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'decodeG:6#%\"eG6 \"F(F(C$-%%subsG6$7&/%%ProdG:F(F(F(F(9\"F(F(/%)SequenceG:F(F(F(F(F1F(F (/%*mu_lengthG%%NULLG/%(EpsilonGF79$7#%\"\"GF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "decode(temp15);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7C%-begin_blockPG%\"ZG%+end_blockPGF$F%F&%.begin_blockLRGF%F%F%F %%,end_blockLRGF$F%F&F$F%F&F$F%F&%.begin_blockRLGF%F%F%%,end_blockRLGF 'F%F%F(F$F%F&"$ }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "seq(count( [S,\{Q\},unlabelled],size=n),n=0..20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "67\"\"!\"\"\"\"\"#\"\"%\"\"*\"#@\"#]\"$?\"\"$*G\"$(p\"%#o\"\"%gS\" %,)*\"&hO#\"&Ar&\"'/z8\"'HHL\"'hP!)\"(]/%>\"(gYo%\")p(48\"" }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 281 20 "Generating functions" }}{PARA 0 "" 0 " " {TEXT -1 84 "A trivariate generating function immediately results fr om the specification. There, " }{XPPEDIT 18 0 "z" "I\"zG6\"" }{TEXT -1 15 " records size, " }{XPPEDIT 18 0 "u" "I\"uG6\"" }{TEXT -1 135 " \+ records the total number of blocks (needed for subsequent permutation \+ enumerations since blocks should be chained to each other), and " } {XPPEDIT 18 0 "v" "I\"vG6\"" }{TEXT -1 29 " records the total length o f " }{XPPEDIT 18 0 "LR" "I#LRG6\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 " RL" "I#RLG6\"" }{TEXT -1 79 " blocks (the number of distiguished excep tions needed for inclusion exclusion)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "gfsolve(\{Q\},unlabelled,z,[[%u,begin_blockP,begin_blo ckLR,begin_blockRL],[v,mu_length]]);" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#<+/-%\"SG6%%\"zG%\"uG%\"vG,$**F)\"\"\"F(F-,(!\"\"F-*&F*F-F(F-F-*(F)F -F(\"\"#F*F-F-F-,,F-F-F0!\"#*&F)F-F(F-F/*&F*F2F(F2F-*(F*F2F(\"\"$F)F-F -F/F//-%,end_blockLRGF'F-/-%\"ZGF'F(/-%,end_blockRLGF'F-/-%.begin_bloc kLRGF'F)/-%-begin_blockPGF'F)/-%.begin_blockRLGF'F)/-%+end_blockPGF'F- /-%*mu_lengthGF'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f_zuv :=subs(\",S(z,u,v));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&f_zuvG,$**% \"uG\"\"\"%\"zGF(,(!\"\"F(*&%\"vGF(F)F(F(*(F'F(F)\"\"#F-F(F(F(,,F(F(F, !\"#*&F'F(F)F(F+*&F-F/F)F/F(*(F-F/F)\"\"$F'F(F(F+F+" }}}{PARA 0 "" 0 " " {TEXT -1 113 " This GF can be checked by comparing its Taylor expans ion and exhaustive listing, at least for small size values." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "map(expand,series(f_zuv,z));" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#+/%\"zG%\"uG\"\"\",&*&F%\"\"\"%\"vGF)F )*$F%\"\"#F)\"\"#,(*&F%F,F*F)F,*&F%F)F*F,F)*$F%\"\"$F)\"\"$,**&F%F,F*F ,F2*&&F%F2F*F)\"\"%*&F%F)F*F2F)*$F%F7F)\"\"%,,*&F%F2F*F,\"\"**&F%F7F*F) \"\"'*&F%F,F*F2F7*$F%\"\"&F)*&F%F)F*F7F)\"\"&-%\"OG6#F)\"\"'" }}} {PARA 0 "" 0 "" {TEXT -1 36 " The exhaustive listing is given by " } {HYPERLNK 17 "Combstruct[allstructs]" 2 "combstruct[allstructs]" "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "map(decode,allstructs([S,\{Q \},unlabelled],size=3));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7&7)%.begi n_blockLRG%\"ZGF&%,end_blockLRG%-begin_blockPGF&%+end_blockPG7)F(F&F)F %F&F&F'7+F(F&F)F(F&F)F(F&F)7'F%F&F&F&F'" }}}{PARA 0 "" 0 "" {TEXT -1 39 "This corresponds to the correct listing" }}{PARA 263 "" 0 "" {TEXT -1 59 " [[1],[2,3]]; [[1],[2],[3]]; [1,2,3]; [[1,2], 3]" }}{PARA 0 "" 0 "" {TEXT -1 19 "hence the monomial " }{XPPEDIT 18 0 "2u^2*v+u*v^2+u^3" ",(*(\"\"#\"\"\"*$%\"uG\"\"#F%%\"vGF%F%*&F'F%*$F) \"\"#F%F%*$F'\"\"$F%" }{TEXT -1 25 " in the series expansion " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 26 "1.2. Avoiding pe'rmutations" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 30 "F rom templates to permutations" }}{PARA 269 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 18 0 "F[n,l]" "&%\"FG6$%\"nG%\"lG" }{TEXT -1 39 " be the numb er of permutations of size " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 28 " distinguished \+ exceptions: " }{XPPEDIT 18 0 "F[n,l]" "&%\"FG6$%\"nG%\"lG" }{TEXT -1 36 " is the number of permutations with " }{XPPEDIT 18 0 "l" "I\"lG6\" " }{TEXT -1 47 " distinguished special successions of the type " } {XPPEDIT 18 0 "(j-1,j)" "6$,&%\"jG\"\"\"\"\"\"!\"\"F$" }{TEXT -1 4 " o r " }{XPPEDIT 18 0 "(j,j-1)" "6$%\"jG,&F#\"\"\"\"\"\"!\"\"" }{TEXT -1 7 ". Let " }{XPPEDIT 18 0 "Q[n]" "&%\"QG6#%\"nG" }{TEXT -1 87 " be th e number of permutations with no exception. Then, by inclusion exclusi on, one has" }}{PARA 266 "" 0 "" {XPPEDIT 18 0 "Q[n]=sum(F[n,l]*(-1)^l ,l=0..n-1)" "/&%\"QG6#%\"nG-%$sumG6$*&&%\"FG6$F&%\"lG\"\"\"),$\"\"\"! \"\"F.F//F.;\"\("!,&F&F/\"\"\"F3" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 154 "(Roughly, one takes objects with at least 0 exception, s ubtract those with at least one exception, then add back those with at least two exceptions, etc.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "F[n,k,l]" "&%\"FG6%%\"nG% \"kG%\"lG" }{TEXT -1 44 " be the number of templates with a total of \+ " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 8 " nodes, " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 36 " blocks, and a total length of the " } {XPPEDIT 18 0 "LR" "I#LRG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "RL" "I#RLG6\"" }{TEXT -1 17 " blocks equal to " }{XPPEDIT 18 0 "l" "I\"lG6 \"" }{TEXT -1 62 ". Then, by looking at all ways to connect the block s, one has" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F[n,l]=sum(F[n,k,l]*phi(k),k)" "/&%\"FG6$%\"n G%\"lG-%$sumG6$*&&F$6%F&%\"kGF'\"\"\"-%$phiG6#F.F/F." }{TEXT -1 6 ", a nd " }{XPPEDIT 18 0 "phi(1)=1" "/-%$phiG6#\"\)"\"\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "phi(k)=(k-2)!" "/-%$phiG6#%\"kG-%*factorialG6#,&F& \"\"\"\"\"#!\"\"" }{TEXT -1 4 " if " }{XPPEDIT 18 0 "k>=2" "1\"\"#%\"k G" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 75 "since any such conne ction is determined by an arbitrary permutation of the " }{XPPEDIT 18 0 "(k-2)" ",&%\"kG\"\"\"\"\"#!\"\"" }{TEXT -1 21 " intermediate blocks ." }}{PARA 0 "" 0 "" {TEXT -1 26 " The trivariate GF of the " } {XPPEDIT 18 0 "F[n,k,l]" "&%\"FG6%%\"nG%\"kG%\"lG" }{TEXT -1 78 " has \+ been determined in the previous section. This shows a chain by which t he " }{XPPEDIT 18 0 "Q[n]" "&%\"QG6#%\"nG" }{TEXT -1 16 " are computab le." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Ob serve that the extension of " }{XPPEDIT 18 0 "phi" "I$phiG6\"" }{TEXT -1 40 " by linearity to an arbitrary series in " }{XPPEDIT 18 0 "u" "I \"uG6\"" }{TEXT -1 12 " is given by" }}{PARA 264 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "phi(h(u))=int(exp(-u)/u^2*(h(u)-(u-u^2)*diff(h(u),u [)[u*=0]),u=0..infinity)" "/-%$phiG6#-%\"hG6#%\"uG-%$intG6$*(-%$expG6#, $F)!\"\"\"\"\"*$F)\"\"#F2,&-F'6#F)F3*&,&F)F3*$F)\"\"#F2F3&-%%diffG6$-F '6#F)&F)6\"6#/F)\"\"!F3F2F3/F);FG%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 46 "That is to say, we just replace in expansions \+ " }{XPPEDIT 18 0 "u-> u^2" ":6#%\"uG7\"6$%)operatorG%&arrowG6\"*$F$\" \"#F)F)" }{TEXT -1 29 " and apply the Euler integral" }}{PARA 265 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "int(exp(-u)*u^k,u=0..infinity)=k! " "/-%$intG6$*&-%$expG6#,$%\"uG!\"\"\"\"\")F+%\"kGF-/F+;\"\"!%)infinit yG-%*factorialG6#F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "T hus, with " }{XPPEDIT 18 0 "F(z,u,v)" "-%\"FG6%%\"zG%\"uG%\"vG" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "sum(F[n,k,l]*z^n*u^k*v^l" "-%$sumG6#* *&%\"FG6%%\"nG%\"kG%\"lG\"\"\")%\"zGF)F,)%\"uGF*F,)%\"vGF+F," }{TEXT -1 2 ", " }}{PARA 268 "" 0 "" {TEXT -1 8 "the OGF " }{XPPEDIT 18 0 "Q( z)=sum(Q[n]*z^n)" "/-%\"QG6#%\"zG-%$sumG6#*&&F$6#%\"nG\"\"\")F&F-F." } {TEXT -1 11 " satisfies " }{XPPEDIT 18+ 0 "Q(z)=phi(F(z,u,-1))" "/-%\"Q G6#%\"zG-%$phiG6#-%\"FG6%F&%\"uG,$\"\"\"!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 271 "" 0 "" {TEXT -1 20 "Generating fun ctions" }}{PARA 0 "" 0 "" {TEXT -1 77 "Start from the previously deter mined trivariate GF and apply the basic chain." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "f_zuv; map(expand,series(f_zuv,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**%\"uG\"\"\"%\"zGF&,(!\"\"F&*&%\"vGF&F'F&F&* (F%F&F'\"\"#F+F&F&F&,,F&F&F*!\"#*&F%F&F'F&F)*&F+F-F'F-F&*(F+F-F'\"\"$F %F&F&F)F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+/%\"zG%\"uG\"\"\",&*&F% \"\"\"%\"vGF)F)*$F%\"\"#F)\"\"#,(*&F%F,F*F)F,*&F%F)F*F,F)*$F%\"\"$F)\" \"$,**&F%F,F*F,F2*&F%F2F*F)\"\"%*&F%F)F*F2F)*$F%F7F)\"\"%,,*&F%F2F*F, \"\"**&F%F7F*F)\"\"'*&F%F,F*F2F7*$F%\"\"&F)*&F%F)F*F7F)\"\"&-%\"OG6#F) \"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 38 "For inclusion-exclusion, one mu st set " }{XPPEDIT 18 0 "v=-1" "/%\"vG,$\"\"\"!\"\"" }{TEXT -1 1 ":" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f_zu:=subs(v=-1,f_zuv);" }} ,{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%f_zuG,$**%\"uG\"\"\"%\"zGF(,(!\"\"F (F)F+*&F'F(F)\"\"#F+F(,,F(F(F)F-*&F'F(F)F(F+*$F)F-F(*&F)\"\"$F'F(F(F+F +" }}}{PARA 0 "" 0 "" {TEXT -1 19 "Application of the " }{XPPEDIT 18 0 "phi" "I$phiG6\"" }{TEXT -1 96 "-transformation (that count the numb er of ways to connect the blocks) requires the modified form" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "k_zu:=normal(f_zu-(u-u^2)*su bs(u=0,diff(f_zu,u)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%k_zuG*,% \"zG\"\"\"%\"uG\"\"#,**&F(F'F&F)F'F&F)*&F(F'F&F'!\"\"F'F'F',&F'F'F&F'F -,*F+F'F&F'F,F-F'F'F-" }}}{PARA 0 "" 0 "" {TEXT -1 15 "The OGF is here " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Q_z:=Int(k_zu*exp(-u)/u^ 2,u=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Q_zG-%$IntG6$* ,%\"zG\"\"\",**&%\"uGF*F)\"\"#F*F)F.*&F-F*F)F*!\"\"F*F*F*,&F*F*F)F*F0, *F,F*F)F*F/F0F*F*F0-%$expG6#,$F-F0F*/F-;\"\"!%)infinityG" }}}{PARA 0 " " 0 "" {TEXT -1 54 "This solves the original counting problem numerica lly:" }}{EXCHG {PARA 0 "> " 0 "" {-MPLTEXT 1 0 119 "Q_ser:=series(map(p roc(e) int(e*exp(-u)/u^2,u=0..infinity) end,map(expand,convert(series( k_zu,z=0,15),polynom))),z,15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&Q _serG+7%\"zG\"\"\"\"\"\"\"\"#\"\"'\"#5\"\"(\"#o\"\")\"$+&\"\"*\"%uT\"# 5\"&u(Q\"#6\"'%e(R\"#7\"([GY%\"#8\")adXa\"#9" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 282 11 "Closed form" }{TEXT -1 16 ". The quantity " }{XPPEDIT 18 0 "Q_z" "I$Q_zG6\"" }{TEXT -1 55 " \+ can be expressed in terms of the exponential integral:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Q_z_closed:=eval(subs(Int=int,Q_z)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+Q_z_closedG**,(%\"zG\"\"\"!\" \"F(*&-%$expG6#*(,&F(F(F'F(F(F'F),&F'F(F)F(F)F(-%#EiG6$F(F.F(F(F(F0F)F 'F(F/F)" }}}{PARA 0 "" 0 "" {TEXT -1 112 "Since one deals with ordinar y generating functions (OGF's), this is to be taken as a formal (asymp totic) series:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "map(normal ,subs(z=1/y,Q_z_closed));map(simplify,asympt(\",y,11));" }}{PARA 11 ". " 1 "" {XPPMATH 20 "6#,$*(,(\"\"\"F&%\"yG!\"\"*(-%$expG6#,$*(,&F'F&F&F &F&F'F&,&F(F&F'F&F(F(F&-%#EiG6$F&F-F&F'F&F&F&F0F(F/F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,0*$%\"yG!\"\"\"\"\"*$F%!\"'\"\"#*$F%!\"(\"#5*$F %!\")\"#o*$F%!\"*\"$+&*$F%!#5\"%uT-%\"OG6#*$F%!#6F'" }}}{PARA 270 "" 0 "" {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 26 "The exponential int egral (" }{XPPEDIT 18 0 "Ei" "I#EiG6\"" }{TEXT -1 46 ") involves the d ivergent series of factorials:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Sum(n!*(-y)^(-n-1),n=0..infinity)=asympt(Ei(1,y)*exp(y),y,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&-%*factorialG6#%\"nG\"\" \"),$%\"yG!\"\",&F+F0F0F,F,/F+;\"\"!%)infinityG,6*$F/F0F,*$F/!\"#F0*$F /!\"$\"\"#*$F/!\"%!\"'*$F/!\"&\"#C*$F/F?!$?\"*$F/!\"(\"$?(*$F/!\")!%S] *$F/!\"*\"&?.%-%\"OG6#*$F/!#5F," }}}{PARA 0 "" 0 "" {TEXT -1 53 "The d ivergent series is also a hypergeometric series:" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 36 "series(hypergeom([1,1],[],z),z=0,9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6/#+7%\"zG\"\"\"\"\"!F%\"\"\"\"\"#\"\"#\"\"'\" \"$\"#C\"\"%\"$?\"\"\"&\"$?(\"\"'\"%S]\"\"(\"&?.%\"\")-%\"OG6#F%\"\"* " }}}{PARA 0 "" 0 "" {TEXT -1 102 "This gives rise to a general conver sion procedure from Exponential integrals to hypergeometric forms:." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "convert_hypergeom:=proc(e) subs(Ei=proc(a,b) if a<>1 then ERROR(`unable to convert`) else exp(-b )/b*hypergeom([1,1],[],-1/b) fi end,e); simplify(\"); end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%2convert_hypergeomG:6#%\"eG6\"F(F(C$-%%sub sG6$/%#EiG:6$%\"aG%\"bGF(F(F(@%09$\"\"\"-%&ERRORG6#%2unable~to~convert G*(-%$expG6#,$9%!\"\"F6F@FA-%*hypergeomG6%7$F6F67\",$*$F@FAFAF6F(F(F5- %)simplifyG6#%\"\"GF(F(" }}}{PARA 0 "" 0 "" {TEXT -1 47 " Hence anothe r closed form for the OGF of the " }{XPPEDIT 18 0 "Q[n]" "&%\"QG6#%\" nG" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Q_z_closed2:=convert_h ypergeom(Q_z_closed);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,Q_z_closed 2G*(%\"zG\"\"\",(F&F'*&F&F'-%*hypergeomG6%7$F'F'7\",$*(0,&F'F'F&F'!\"\" F&F',&F&F'F2F'F'F2F'F'F'F'F'F1!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 283 11 "Recurrences" }{TEXT -1 89 ". Holonomic descriptions (by means of linear differential equations) can be obtai ned by " }{HYPERLNK 17 "Gfun[holexprtodiffeq]" 2 "gfun[holexprtodiffeq ]" "" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Qz_o de0:=holexprtodiffeq(Q_z_closed2,Y(z));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(Qz_ode0G,6*&,.*$%\"zG\"\"%\"\"\"*$F)\"\"&F+*$F)\"\"$F*!\"\"F+ F)F0*$F)\"\"#F*F+-%\"YG6#F)F+F+*&,(F(!\"#F1F+*$F)\"\"'F+F+-%%diffG6$F3 F)F+F+F(F0*&F)F/&%#_CG6#\"\"!F+F8F.F8*&F)F*F?F+F0*&F?F+F)F2F+F,F0F)F+F 1F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "subs(Y(z)=Qz_ser,Qz_ ode0):series(\",z=0,11);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+/%\"zG,$% 'Qz_serG!\"\"\"\"!,&F&F'\"\"\"F*\"\"\",(F&\"\"%F'F*&%#_CG6#\"\"!F*\"\" #,(F&F-F.!\"#F4F*\"\"$,(F.F'F'F*F&F*\"\"%,&F'F*F&F*\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Qz_ode:=subs(_C[0]=1,Qz_ode0);" }} {PARA 11 "" 1 1"" {XPPMATH 20 "6#>%'Qz_odeG,.*&,.*$%\"zG\"\"%\"\"\"*$F) \"\"&F+*$F)\"\"$F*!\"\"F+F)F0*$F)\"\"#F*F+-%\"YG6#F)F+F+*&,(F(!\"#F1F+ *$F)\"\"'F+F+-%%diffG6$F3F)F+F+F(F8F.!\"%F,F0F)F+" }}}{PARA 0 "" 0 "" {TEXT -1 58 " The transformation to a linear recurrence is obtained by " }{HYPERLNK 17 "Gfun[diffeqtorec]" 2 "gfun[diffeqtorec]" "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Q_rec:=diffeqtorec(\{Qz_ode= 0,Y(0)=0\},Y(z),u(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&Q_recG<)/ -%\"uG6#\"\"!F*/-F(6#\"\"\"F.,.*&,&%\"nGF.F.F.F.-F(6#F2F.F.-F(6#F1F.*& F2F.-F(6#,&F2F.\"\"#F.F.!\"#-F(6#,&\"\"$F.F2F.\"\"%*&F?F.-F(6#,&F2F.FA F.F.F.-F(6#,&F2F.\"\"&F.!\"\"/-F(6#F;F*/-F(6#F@F*/-F(6#FAF*/-F(6#FIF* " }}}{PARA 0 "" 0 "" {TEXT -1 106 "This provides an algorithm that use s a linear number of arithmetic operations to determine the quantities " }{XPPEDIT 18 0 "Q[n]" "&%\"QG6#%\"nG" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "ha:=rectoproc(Q_rec,u(n),remember); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#haG:6#%\"nG62\"6#%)rememberGE\\s '\"\"#\"\"!\"\"&F-F-F-\"\"\"F/\"\"%F-\"\"$F-,.-9!6#,&9$F/!\"&F/!\"%-F4 6#,&F7F/!\"#F/F0-F46#,&F7F/F9F/F/-F46#,&F7F/!\"$F/\"#5-F46#,&F7F/!\"\" F/F=*&,(FAF=F3F/FFF/F/F7F/F/F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ha(5);ha(50);ha(500);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"hn'o([)*fvbLMyJ>YS\\* o(HYe&=]7]D*yi2h\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#\"cao'4POI_T[PIj TXsi2[CFn9$Qf^k(4O_ee`#Hl_:)HdQWfzrpncPks%>#)=o')*f+*HZHlP<**>R;PdfHcF +t\"ph+z:?R)Hj:k0QGy\"e/XoHE3h%)Qlw!yzb-I4U=:bt$Q\"4$>Z@hfx7>@kV1%fke647aaG \"piqels3of+u\\E\"))>VSKNX&3Oc#o9%RgWKG>L;ce%z$)y/#y;QkBX=5`-VzdkfIwVV /@g5Q%=2$3Qz*G$Gfn))\\%)e/ " 0 "" {MPLTEXT 1 0 30 "seq(ha(n)/(n-2)! *1.0,n=2..50);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6S\"\"!F#F#F#$\"+LLLL$ )!#6F$$\"+WWWW%*F&$\"+@\\j?**F&$\"+a#=_.\"!#5$\"+'\\2&o5F-$\"+@\\j&4\" F-$\"+=v.=6F-$\"+V&fo8\"F-$\"+NN*G:\"F-$\"+ahrm6F-$\"+9`vy6F-$\"+t_L*= \"F-$\"+$=1()>\"F-$\"+6T127F-$\"+/\\c97F-$\"+$*RL@7F-$\"+vMZF7F-$\"+%H nIB\"F-$\"+.^=Q7F-$\"+(=&)GC\"F-$\"+&o;sC\"F-$\"+>8A^7F-$\"+WZ$\\D\"F- $\"+6wQe7F-$\"+9kgh7F-$\"+uThk7F-$\"+()4Vn7F-$\"+yW2q7F-$\"+m,cs7F-$\" +s8\"H\"F-$\"+s+`#H\"F -$\"+en'QH\"F-$\"+bf9&H\"F-$\"+'HrjH\"F-$\"+5ha(H\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ha(500)/(498)!*1.0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+MJ#zM\"!#5" }}}{EXCHG {P4ARA 0 "> " 0 "" {MPLTEXT 1 0 18 "exp(-2)=exp(-2.0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%$expG6#!\"#$\"+KGN`8!#5" }}}{PARA 0 "" 0 "" {TEXT -1 63 "And one c an even guess the next term in an asymptotic expansion" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "seq(n*(1-ha(n)/(n-2)!*exp(2.0)),n=6 ..50);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6O$\"+^>Z0B!\"*$\"+wsr*o#F%$\" +9w:@:BF%$\"+E0:&G#F%$\"+xaQgAF%$\"+ %[`&RAF%$\"+\"\\2=A#F%$\"+=-^1AF%$\"+f%)=$>#F%$\"+0H[\"=#F%$\"+fi6r@F% $\"+Q6(=;#F%$\"+bZd`@F%$\"+U\")3Y@F%$\"+c\")HR@F%$\"+==6L@F%$\"+L?XF@F %$\"+NUDA@F%$\"+dSY<@F%$\"+]`.8@F%$\"+]'G*3@F%$\"+W+60@F%$\"+R-b,@F%$ \"+gPA)4#F%$\"+1%3^4#F%$\"+pY=#4#F%$\"+/aV*3#F%$\"+Ia%o3#F%$\"+,8S%3#F %$\"+**44#3#F%$\"+$)Q!*z?F%$\"+!GIy2#F%$\"+W;'e2#F%$\"+[,*R2#F%$\"+3)3 A2#F%$\"+@7^q?F%$\"+b;*)o?F%$\"+D[Mn?F%$\"+Bf'e1#F%$\"+M1Xk?F%$\"+o[4j ?F%$\"+5]zh?F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "nn:=500; \+ ha(nn)/((nn-2)!*(1.0-2.0/nn)*exp(-2.0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#nnG\5"$+&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+u\"z)****!#5 " }}}{PARA 0 "" 0 "" {TEXT -1 42 "Thus, with reasonable certainty, we \+ expect" }}{PARA 272 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Q[n]/(n-2)! =exp(-2)*(1-2/n+O(1/n^2))" "/*&&%\"QG6#%\"nG\"\"\"-%*factorialG6#,&F'F (\"\"#!\"\"F.*&-%$expG6#,$\"\"#F.F(,(\"\"\"F(*&\"\"#F(F'F.F.-%\"OG6#*& \"\"\"F(*$F'\"\"#F.F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 72 "The principle of a proof based on the generating function method i s that" }}{PARA 273 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "coeff[z^n]* hypergeom([1,1],[],z+d*z^2+O(z^3))=n!*e^d*(1+o(1))" "/*&&%&coeffG6#)% \"zG%\"nG\"\"\"-%*hypergeomG6%7$\"\"\"\"\"\"7\",(F(F**&%\"dGF**$F(\"\" #F*F*-%\"OG6#*$F(\"\"$F*F**(-%*factorialG6#F)F*)%\"eGF4F*,&\"\"\"F*-% \"oG6#\"\"\"F*F*" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 108 "prov ided that the argument of the hypergeometric is a function that is ana lytic at the origin. Here, one has" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Q_z_closed2;" }}{PARA 11 "" 1 "6" {XPPMATH 20 "6#*(%\" zG\"\"\",(F$F%*&F$F%-%*hypergeomG6%7$F%F%7\",$*(,&F%F%F$F%!\"\"F$F%,&F $F%F0F%F%F0F%F%F%F%F%F/!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "-z*(z-1)/(1+z)=series(-z*(z-1)/(1+z),z=0,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*(,&\"\"\"F'%\"zGF'!\"\"F(F',&F(F'F)F'F'F)+/F(F'\"\" \"!\"#\"\"#\"\"#\"\"$F-\"\"%F/\"\"&-%\"OG6#F'\"\"'" }}}{PARA 0 "" 0 " " {TEXT -1 59 "so that the asymptotic proportion of legal permutations is " }{TEXT 284 8 "provedly" }{TEXT -1 10 " equal to " }{XPPEDIT 18 0 "e^(-2)" ")%\"eG,$\"\"#!\"\"" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Qn_asy:=`Q `[n]/(n-2)!=exp(-2)*(1+O(1/n));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Qn_asyG/*&&%#Q~G6#%\"nG\"\"\"-%*fac torialG6#,&F*F+!\"#F+!\"\"*&-%$expG6#F0F+,&F+F+-%\"OG6#*$F*F1F+F+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 " Results" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "We have thus obtained here a few original results regarding the e numeration7 of avoiding permutations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 275 "" 0 "" {TEXT 291 9 "Theorem 1" }{TEXT -1 2 ". " }{TEXT 292 11 "The number " }{XPPEDIT 293 0 "Q[n]" "&%\"QG6#%\"nG" }{TEXT 294 59 " of avoiding permutations has ordinary generating function:" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Q_z_closed=Q_z_closed2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/**,(%\"zG\"\"\"!\"\"F'*&-%$expG6#*(,& F'F'F&F'F'F&F(,&F&F'F(F'F(F'-%#EiG6$F'F-F'F'F'F/F(F&F'F.F(*(F&F',(F&F' *&F&F'-%*hypergeomG6%7$F'F'7\",$*(F.F(F&F'F/F'F(F'F'F'F'F'F.!\"#" }}} {PARA 276 "" 0 "" {TEXT -1 17 "The coefficients " }{XPPEDIT 18 0 "Q[n] " "&%\"QG6#%\"nG" }{TEXT -1 24 " satisfy the recurrence:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Q_rec;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<)/-%\"uG6#\"\"!F(/-F&6#\"\"\"F,,.*&,&%\"nGF,F,F,F,-F&6#F0F,F,-F &6#F/F,*&F0F,-F&6#,&F0F,\"\"#F,F,!\"#-F&6#,&\"\"$F,F0F,\"\"%*&F=F,-F&6 #,&F0F,F?F,F,F,-F&6#,&F0F,\"\"&F,!\"\"/-F&6#F9F(/-F&6#F>F(/-F&6#F?F(/- F&6#FGF(" }}}{PARA 0 "" 0 "" {TEXT 295 85 "Ac8cordingly, the ordinary g enerating function satisfies the corresponding linear ODE:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "map(factor,Qz_ode);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*(,&\"\"\"F&%\"zGF&F&,(*$F'\"\"%F&*$F'\"\"#F*!\" \"F&F&-%\"YG6#F'F&F&**F'F,,&F'F&F-F&F,F%F,-%%diffG6$F.F'F&F&F)!\"#*$F' \"\"$!\"%*$F'\"\"&F-F'F&" }}}{PARA 277 "" 0 "" {TEXT -1 17 "The coeffi cients " }{XPPEDIT 18 0 "Q[n]" "&%\"QG6#%\"nG" }{TEXT -1 32 " satisfy \+ the asymptotic estimate" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Qn _asy;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#Q~G6#%\"nG\"\"\"-%*fact orialG6#,&F(F)!\"#F)!\"\"*&-%$expG6#F.F),&F)F)-%\"OG6#*$F(F/F)F)" }}}} }{SECT 0 {PARA 4 "" 0 "" {TEXT 285 25 "2. Paths with outer nodes" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " We now d efine " }{TEXT 297 14 "avoiding paths" }{TEXT -1 27 ". An avoiding pat h of size " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 38 " is a sequence of values from the set " }{XPPEDIT 18 0 "\{1..n\} union \{-1\}" "-%&u n9ionG6$<#;\"\"\"%\"nG<#,$\"\"\"!\"\"" }{TEXT -1 8 ", where " } {XPPEDIT 18 0 "-1" ",$\"\"\"!\"\"" }{TEXT -1 68 " represents generical ly a node that does not belong to the interval " }{XPPEDIT 18 0 "[1,`. .`,n]" "7%\"\"\"%#..G%\"nG" }{TEXT -1 64 ", with the constraint that t he initial value of the sequence is " }{XPPEDIT 18 0 "1" "\"\"\"" } {TEXT -1 21 ", the final value is " }{XPPEDIT 18 0 "n" "I\"nG6\"" } {TEXT -1 48 ", no successive value pairs can be of the type " } {XPPEDIT 18 0 "j,j+1" "6$%\"jG,&F#\"\"\"\"\"\"F%" }{TEXT -1 4 " or " } {XPPEDIT 18 0 "j,j-1" "6$%\"jG,&F#\"\"\"\"\"\"!\"\"" }{TEXT -1 20 ", a nd no value from " }{XPPEDIT 18 0 "[1,`..`,n]" "7%\"\"\"%#..G%\"nG" } {TEXT -1 73 " can occur more than once. Here are the sets of avoiding \+ paths for sizes " }{XPPEDIT 18 0 "n=2,3,4,5" "6&/%\"nG\"\"#\"\"$\"\"% \"\"&" }{TEXT -1 1 ":" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"#<\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"$<#7%\"\"\"!\"\"F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"%<%7&\"\"\"!\"\"F'F#7&F&:F'\"\"#F#7&F&\"\"$F'F# " }}{PARA 0 "" 0 "" {XPPMATH 20 "6$\"\"& " 0 "" {MPLTEXT 1 0 45 "sp0:=S=Prod(Sequence( Prod(a,Sequence(b))),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sp0G/% \"SG-%%ProdG6$-%)SequenceG6#-F(6$%\"aG-F+6#%\"bG%\"xG" }}}{PARA 0 "" 0 "" {TEXT -1 69 "(It suffices to decompose according to each occurren ce of the letter " }{XPPEDIT 18 0 "a" "I\"aG6<\"" }{TEXT -1 3 ".) " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "We first \+ need so-called \"outer points\" that are taken from outer space." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Outerpoints:=Sequence(Prod(Z ,mu_outerpoint));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,OuterpointsG-% )SequenceG6#-%%ProdG6$%\"ZG%.mu_outerpointG" }}}{PARA 0 "" 0 "" {TEXT -1 28 "We also need \"inner points\"." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Innerpoints:=Sequence(Prod(Z,mu_innerpoint));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%,InnerpointsG-%)SequenceG6#-%%ProdG6 $%\"ZG%.mu_innerpointG" }}}{PARA 0 "" 0 "" {TEXT -1 194 "Size is defin ed as the cumulated number of points in the pair of paths that underli es an avoiding path in the sense above: it is thus equal to the length of the avoiding path plus the number of " }{XPPEDIT 18 0 "-1" ",$\"\" \"!\"\"" }{TEXT -1 42 " symbols corresponding to the outer nodes." } {TEXT -1 101 " We thus introduce a special notation for nodes of the i nteger line= that are shared by the two paths:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Z2:=Prod(Z,Z);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#Z2G-%%ProdG6$%\"ZGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Now, the three types of blocks are described by" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 218 "sp1:=Prod(mu_block,Z2,Out erpoints,Innerpoints); sp2:=Prod(mu_block,Z2,Sequence(Prod(mu_length,Z 2),card>=1),Outerpoints,Innerpoints);\nsp3:=Prod(Sequence(Prod(mu_leng th,Z2),card>=1),Z2,Outerpoints,Innerpoints,mu_block);\n" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$sp1G-%%ProdG6&%)mu_blockG-F&6$%\"ZGF+-%)Seque nceG6#-F&6$F+%.mu_outerpointG-F-6#-F&6$F+%.mu_innerpointG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$sp2G-%%ProdG6'%)mu_blockG-F&6$%\"ZGF+-%)Sequ enceG6$-F&6$%*mu_lengthGF)1\"\"\"%%cardG-F-6#-F&6$F+%.mu_outerpointG-F -6#-F&6$F+%.mu_innerpointG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$sp3G- %%ProdG6'-%)SequenceG6$-F&6$%*mu_lengthG-F&6$%\"ZGF01\"\"\"%%cardGF.-F )6#-F&6$F0%.mu_outerpointG-F)6#-F&6$F0%>.mu_innerpointG%)mu_blockG" }}} {PARA 0 "" 0 "" {TEXT -1 10 "(Clearly, " }{XPPEDIT 18 0 "sp2" "I$sp2G6 \"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sp3" "I$sp3G6\"" }{TEXT -1 34 " are combinatorially isomorphic.) " }}{PARA 0 "" 0 "" {TEXT -1 49 "Th e blocks that can occur at the end are of type " }{XPPEDIT 18 0 "x" "I \"xG6\"" }{TEXT -1 25 " and can only be of type " }{XPPEDIT 18 0 "sp1 " "I$sp1G6\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "sp2" "I$sp2G6\"" } {TEXT -1 42 " but without outer points nor innerpoints." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sp1x:=Prod(mu_block,Z2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sp1xG-%%ProdG6$%)mu_blockG-F&6$%\"ZGF+" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "sp2x:=Prod(mu_block,Z2,Seq uence(Prod(mu_length,Z2),card>=1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%%sp2xG-%%ProdG6%%)mu_blockG-F&6$%\"ZGF+-%)SequenceG6$-F&6$%*mu_leng thGF)1\"\"\"%%cardG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 21 "Then, the grammar is:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEX?T 1 0 138 "Q:=subs([a=Union(sp1,sp2),b=sp3,x=Union(sp1x,sp2x)], sp0), mu_block=Epsilon, mu_length=Epsilon,mu_outerpoint=Epsilon,mu_inn erpoint=Epsilon;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"QG6'/%\"SG-%%P rodG6$-%)SequenceG6#-F)6$-%&UnionG6$-F)6&%)mu_blockG-F)6$%\"ZGF8-F,6#- F)6$F8%.mu_outerpointG-F,6#-F)6$F8%.mu_innerpointG-F)6'F5F6-F,6$-F)6$% *mu_lengthGF61\"\"\"%%cardGF9F>-F,6#-F)6'FEF6F9F>F5-F16$-F)6$F5F6-F)6% F5F6FE/F5%(EpsilonG/FIFX/F=FX/FBFX" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "draw([S,\{Q\},unlabelled],size=8);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#-%%ProdG6$-%)SequenceG6$-F$6$-F$6&%)mu_blockG-F$6$%\" ZGF0%(EpsilonGF1F1F)-F$6%F-F.-F'6#-F$6$%*mu_lengthGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "seq(count([S,\{Q\},unlabelled],size =n),n=0..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "67\"\"!F#\"\"\"F#\"\"#F %\"\"(\"#7\"#I\"#i\"$V\"\"$7$\"$-(\"%a:\"%rM\"%;x\"&)><\"&y#Q\"&j_)\"' S)*=\"'#yA%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "for j from 0 to 6 do allstructs([S,\{Q\},unlabelled],siz@e=j) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#-%%ProdG6$%(EpsilonG-F%6$%)mu_blockG-F%6$% \"ZGF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7$-%%ProdG6$-%)SequenceG6#-F%6$-F%6&%)mu_blockG-F%6$%\" ZGF1%(EpsilonGF2F2-F%6$F.F/-F%6$F2-F%6%F.F/-F(6#-F%6$%*mu_lengthGF/" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#7$-%%ProdG6$-%)SequenceG6#-F%6$-F%6&% )mu_blockG-F%6$%\"ZGF1%(EpsilonG-F(6#-F%6$F1%.mu_innerpointGF2-F%6$F.F /-F%6$-F(6#-F%6$-F%6&F.F/-F(6#-F%6$F1%.mu_outerpointGF2F2F8" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7)-%%ProdG6$-%)SequenceG6#-F%6$-F%6&%)mu_blo ckG-F%6$%\"ZGF1-F(6#-F%6$F1%.mu_outerpointG-F(6#-F%6$F1%.mu_innerpoint G%(EpsilonG-F%6$F.F/-F%6$-F(6#-F%6$-F%6&F.F/F<-F(6$F9F9F " 0 "" {MPLTEXT 1 0 93 "gfsolve(\{Q\},unlabelle d,z,[[u,mu_block],[v,mu_length],[w1,mu_outerpoint],[w2,mu_innerpoint]] );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<(/-%.mu_innerpointG6'%\"zG%\"uG %\"vG%#w1G%#w2GF,/-%\"ZGF'F(/-%)mu_blockGF'F)/-%.mu_outerpointGF'F+/-% *mu_lengthGF'F*/-%\"SGF',$**F)\"\"\"F(\"\"#,4!\"\"F>*&F(F>F,F>F>*&F(F> F+F>F>*(F(F?F+F>F,F>FA*&F*F>F(F?F>*(F*F>F(\"\"$F,F>FA*(F*F>F(FGF+F>FA* *F*F>F(\"\"%F+F>F,F>F>*(F)F>F(FJF*F>F>F>,>F>F>FBFAFCFAFDF>FE!\"#FFF?FH F?FIFM*&F)F>F(F?FA*&F*F?F(FJF>*(F*F?F(\"\"&F,F>FA*(F*F?F(FQF+F>FA**F*F ?F(\"\"'F+F>F,F>F>*(F*F?F(FTF)F>F>FAFA" }}}{PARA 0 "" 0 "" {TEXT -1 44 " For inclusion-exclusion, one must set v=-1:" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 48 "f_zu:=factor(subs(v=-1,subs(\",S(z,u,v,w1,w2)) ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%f_zuG*,%\"uG\"\"\"%\"zG\"\"# ,4F'F'*&F(F'%#w2GF'!\"\"*&F(F'%B#w1GF'F-*(F(F)F/F'F,F'F'*$F(F)F'*&F(\" \"$F,F'F-*&F(F3F/F'F-*(F(\"\"%F/F'F,F'F'*&F&F'F(F6F'F',&F'F'F1F'F-,6F5 F'F7F'F2F-F4F-F0F'F1F'*&F&F'F(F)F-F+F-F.F-F'F'F-" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 19 "Application of the " } {XPPEDIT 18 0 "phi" "I$phiG6\"" }{TEXT -1 97 "-transformation (that co unts the number of ways to connect the blocks) requires the modified f orm" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "h_zu:=factor(normal(f _zu-(u-u^2)*subs(u=0,diff(f_zu,u))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%h_zuG*,%\"uG\"\"#%\"zGF',6\"\"\"F**$F(F'F'*&F(\"\"$%#w1GF*!\"\"* &F&F*F(\"\"%F**(F(F1F.F*%#w2GF*F**&F(F-F3F*F/*&F&F*F(F'F/*&F(F*F3F*F/* &F(F*F.F*F/*(F(F'F.F*F3F*F*F*,&F*F*F+F*F/,6F2F*F0F*F4F/F,F/F8F*F+F*F5F /F6F/F7F/F*F*F/" }}}{PARA 0 "" 0 "" {TEXT -1 42 "This solves the origi nal counting problem:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "Q_ ser:=series(map(proc(e) int(e*exp(-u)/u^2,u=0..infinity) end,map(expan d,convert(series(h_zu,z=0,12),polynom))),z,15);" }}{PARA 12 "C" 1 "" {XPPMATH 20 "6#>%&Q_serG+3%\"zG\"\"\"\"\"#,&%#w2GF'%#w1GF'\"\"&,(*$F+ \"\"#F'*&F*F'F+F'F'*$F*F/F'\"\"',**$F+\"\"$F'*&F*F/F+F'F'*&F*F'F+F/F'* $F*F5F'\"\"(,2*$F+\"\"%F'F1F'*&F*F/F+F/F'F0F/*&F+F5F*F'F'F.F'*$F*FFL*&F*FGF+F'F'F 0FG*$F+FLF'F;F5F=\"\"(*&F*F " 0 "" {MPLTEXT 1 0 41 "Q_z:=Int(h_zu*exp(-u)/u^2,u=0..infi nity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Q_zG-%$IntG6$*,%\"zG\"\"# ,6\"\"\"F,*$F)F*F**&F)\"\"$%#w1GF,!\"\"*&%\"uGF,F)\"\"%F,*(F)F4F0F,%#w 2GF,F,*&F)F/F6F,F1*&F3F,F)F*F1*&F)F,F6F,F1*&F)F,F0F,F1*(F)F*F0F,F6F,F, F,,&F,F,F-F,F1,6F5F,F2F,F7F1F.F1F;F,F-F,F8F1F9F1F:F1F,F,F1-%$expG6#,$F 3F1F,/F3;\"\"!%)infinityG" }}}{PARA 0 "" 0 "" {TEXT -1 63 "And this ca n be expressed in terms of the exponential integral:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Q_z_closed:=eval(subs(Int=int,Q_z));" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%+Q_z_closedG*,%\"zG\"\"#,(*&F&F'-%$e xpG6#*,,&\"\"\"F/*$F&F'F/F/,&%#w2GF/%#w1GF/F/F&!\"\",&F&F/F4F/F4,&F/F/ F&F/F4F/F/*&-%#EiG6$F/*(,2*(F&\"\"%F3F/F2F/F/*&F&\"\"$F2F/F4*&F&F@F3F/ F4*(F&F'F3F/F2F/F/F0F/*&F&F/F2F/F4*&F&F/F3F/F4F/F/F/F&!\"#,&F0F/F4F/F4 F/-F+6#*,F.F/,&FBF/F/F/F/F&FEF5F4F6F4F/F/F*F4F/FFF4F.F4-F+6#,$F-F4F/" }}}{PARA 0 "" 0 "" {TEXT -1 57 "Again, there is Ean \"explicit form\" o f the OGF the problem" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Qz_ closed2:=map(factor,convert_hypergeom(Q_z_closed));" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%+Qz_closed2G*,,4*(%\"zG\"\"%%#w1G\"\"\"%#w2GF+F+*&F (\"\"$F,F+!\"\"*&F(F.F*F+F/*&F(\"\"#-%*hypergeomG6%7$F+F+7\",$*.F(F2,& F(F+F/F+F+,&F+F+F(F+F+,&F+F+*$F(F2F+F/,&*&F(F+F,F+F+F/F+F/,&*&F(F+F*F+ F+F/F+F/F/F+F+F=F+*(F(F2F*F+F,F+F+F?F/FAF/F+F+F+F(F2FF/F@F/" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "Qz_closed3:=factor(Qz_close d2-subs(hypergeom=0,Qz_closed2))+factor(subs(hypergeom=0,Qz_closed2)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+Qz_closed3G,&*,%\"zG\"\"%-%*hyp ergeomG6%7$\"\"\"F-7\",$*.F'\"\"#,&F'F-!\"\"F-F-,&F-F-F'F-F-,&F-F-*$F' F1F-F3,&*&F'F-%#w2GF-F-F3F-F3,&*&F'F-%#w1GF-F-F3F-F3F3F-F5!\"#F7F3F:F3 F-*&F5F3F'F1F-" }}}{PARA 0 "" 0 "" {TEXT -1 20 "The diagonal, where " }{XPPEDIT 18 0 "w1" "I#w1G6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "w2 " "I#w2G6\"" }{TEXT -1 55 " appear with equal exponents, is then easil y exFtracted." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "Q_order:=20 : Qz_ser:=map(normal,series(Qz_closed3,z=0,Q_order+2)):subs([w1=w,w2=t /w],Qz_ser): subs([w1=w,w2=t/w],Qz_ser): map(series,\",w=0,2*Q_order+5 ):Qzt_ser:=map(coeff,\",w,0);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(Qz t_serG+7%\"zG\"\"\"\"\"#%\"tG\"\"',&*$F)\"\"#F'F)F-\"\"),(*$F)\"\"$F'F )\"\"&F,\"\"(\"#5,,F)\"#?F,\"#R*$F)\"\"%F'F0\"#9F-F'\"#7,.F0\"$R\"F)\" $:\"*$F)F2F'F8\"#B\"#5F'F,\"$X#\"#9,0F8\"$`$F)\"$!zF0\"%S8*$F)\"\"'F'F ,\"%H=F?\"#M\"#oF'\"#;,2F?\"$T(*$F)F3F'F8\"%r[F,\"&rd\"FH\"#ZF0\"&\\P \"\"$+&F'F)\"% " 0 "" {MPLTEXT 1 0 36 "subs(t=0,Qzt_ ser);subs(t=1,Qzt_ser);" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#+1%\"zG\"\"\"\"\"#\"\"#\"#7\"#5\"#9\"#o\"#;\"$+& \"#=\"%uT\"#?-%\"OG6#F%\"#A" }}{GPARA 11 "" 1 "" {XPPMATH 20 "6#+7%\"zG \"\"\"\"\"#F%\"\"'\"\"$\"\")\"#8\"#5\"#w\"#7\"$L&\"#9\"%:W\"#;\"&(*=% \"#=\"'GxW\"#?-%\"OG6#F%\"#A" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 25 " 2.3. Exhaustive listings" }}{PARA 0 "" 0 "" {TEXT -1 227 "Here's a p rocedure that lists balanced avoiding pairs exhaustively (by generati ng all permutations and filtering the relevant ones) and derives the c ounting polynomials. This fully confirms the GF computations done prev iously." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 576 "test:=proc() \nl ocal i, perms, dontcares, poly, count1, p, p1, j, x, s, n, listing; n: =args[1];\nperms:=combstruct[allstructs](Permutation([seq(i,i=2..n-1)] ),size=n-2); dontcares := combstruct[allstructs](Subset(\{seq(i,i=2..n -1)\}));\nlisting:=\{\};\npoly:=0;\nfor s in dontcares do\ncount1:=0; \nfor p in perms do p1:=[1,op(p),n]; \nfor x in s do p1[x]:=-1; od;\nf or j from 1 to n-1 while abs(p1[j+1]-p1[j])<>1 do \nod; \nif j=n then \+ count1:=count1+1; listing:=listing union \{p1\} fi;\nod;\nHpoly:=poly+c ount1*t^nops(s)/nops(s)!;\nod;\nif nargs>1 then RETURN(sort(listing)) \+ fi;\nRETURN(poly);\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%testG:6 \"6.%\"iG%&permsG%*dontcaresG%%polyG%'count1G%\"pG%#p1G%\"jG%\"xG%\"sG %\"nG%(listingGF&F&C*>8.&9\"6#\"\"\">8%-&%+combstructG6#%+allstructsG6 $-%,PermutationG6#7#-%$seqG6$8$/FJ;\"\"#,&F6F:!\"\"F:/%%sizeG,&F6F:!\" #F:>8&-F>6#-%'SubsetG6#<#FG>8/<\">8'\"\"!?&8-FU%%trueGC%>8(F[o?&8)F8*7%F:-%#opG6#FcoF6?&8,F]oF^o>&Ffo6#F\\pFO?(8+F:F:FN0-%$absG6#,&&F fo6#,&FapF:F:F:F:&Ffo6#FapFOF:F&@$/FapF6C$>Fao,&FaoF:F:F:>Fgn-%&unionG 6$Fgn<#Ffo>Fjn,&FjnF:*(FaoF:)%\"tG-%%nopsG6#F]oF:-%*factorialG6#F[rFOF :@$2F:9#-%'RETURNG6#-%%sortG6#Fgn-Fer6#FjnF&F&" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 34 "for j from 2 to 6 do j,test(j) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$\"\"$%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"%,&*$%\"tG\"\"#\" \"\"F&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"&,(*$%\"tG\"\"$\"\"\"F &F#*$F&\"\"#\I"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"',,%\"tG\"#?* $F%\"\"#\"#R*$F%\"\"%\"\"\"*$F%\"\"$\"#9F(F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "for j from 2 to 6 do j,test(j,LIST_THEM_ALL) od; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"#<\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"$<#7%\"\"\"!\"\"F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"%<%7&\"\"\"\"\"$!\"\"F#7&F&F(F(F#7&F&F(\"\"#F#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$\"\"& " 0 "" {MPLTEXT 1 0 37 "remove(has,test(6,LIST_THE M_ALL),-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$7(\"\"\"\"\"%\"\"#\" \"&\"\"$\"\"'7(F%F)F(F'F&F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "remove(has,test(7,LIST_THEM_ALL),-1);" }}{PARA 12 "" 1 "" {XPPMATH 20K "6#<,7)\"\"\"\"\"$\"\"&\"\"#\"\"'\"\"%\"\"(7)F%F&F)F*F(F'F +7)F%F*F(F)F&F'F+7)F%F*F)F(F'F&F+7)F%F*F)F&F'F(F+7)F%F'F(F*F)F&F+7)F%F 'F&F)F(F*F+7)F%F'F&F)F*F(F+7)F%F)F&F'F(F*F+7)F%F)F*F(F'F&F+" }}}{PARA 0 "" 0 "" {TEXT -1 68 "These results confirm the GF computation of the previous subsection." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 32 " 2.4. E xplicit binomial formulae" }}{PARA 0 "" 0 "" {TEXT -1 91 "We can now r eturn to the GF of avoiding paths enumerated by size and number of out er nodes." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Qz_closed3;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*,%\"zG\"\"%-%*hypergeomG6%7$\"\"\"F +7\",$*.F%\"\"#,&F%F+!\"\"F+F+,&F+F+F%F+F+,&F+F+*$F%F/F+F1,&*&F%F+%#w2 GF+F+F1F+F1,&*&F%F+%#w1GF+F+F1F+F1F1F+F3!\"#F5F1F8F1F+*&F3F1F%F/F+" }} }{PARA 0 "" 0 "" {TEXT -1 16 "The coefficient " }{XPPEDIT 18 0 "c(n,j, k) " "-%\"cG6%%\"nG%\"jG%\"kG" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "z^n* (w1^j*w2^k" "*&)%\"zG%\"nG\"\"\"*&)%#w1G%\"jGF&)%#w2G%\"kGF&F&" } {TEXT -1 77 " is obtained by straight expaLnsion and avoiding paths are then enumerated by " }{XPPEDIT 18 0 "C(n,j)=c(n,j,j)" "/-%\"CG6$%\"nG %\"jG-%\"cG6%F&F'F'" }{TEXT -1 103 ". The corresponding formulae are o btained directly by symbolic expansions (performed manually, though!) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "j:='j': C_formula:=C(n+ 2,j)=Sum(Sum( (-1)^(k1+k2)*(n-j-k1-k2)!*'B'(n-j-k1-k2,k1)* 'B'(n-j-k1+ 1,k2)*'B'(n-k1-k2,j)^2, k1=0..n-j-k2), k2=0..n-j);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%*C_formulaG/-%\"CG6$,&%\"nG\"\"\"\"\"#F+%\"jG-%$Sum G6$-F/6$*,)!\"\",&%#k1GF+%#k2GF+F+-%*factorialG6#,*F*F+F-F5F7F5F8F5F+- %\"BG6$F6$,*F*F+F-F5F7F5F+F+F8F+-F>6$,(F*F+F7F5F8F5F-F,/F7;\" \"!,(F*F+F-F5F8F5/F8;FH,&F*F+F-F5" }}}{PARA 0 "" 0 "" {TEXT -1 6 "wher e " }{XPPEDIT 18 0 "B(a,b)" "-%\"BG6$%\"aG%\"bG" }{TEXT -1 30 " is the binomial, coefficient " }{XPPEDIT 18 0 "B(a,b)=a!/b!/(a-b)!" "/-%\"BG 6$%\"aG%\"bG*(-%*factorialG6#F&\"\"\"-F*6#F'!\"\"-F*6#,&F&F,F'F/F/" } {TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Bin:=proc(n ,Mk) if n<0 or k<0 then 0 else binomial(n,k) fi; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$BinG:6$%\"nG%\"kG6\"F)F)@%529$\"\"!29%F.F.-%)bino mialG6$F-F0F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "coef:=p roc(n,j) option remember; local k1,k2;\nadd( add( (-1)^(k1+k2)*(n-j-k1 -k2)!*Bin(n-j-k1-k2,k1)* Bin(n-j-k1+1,k2)*Bin(n-k1-k2,j)^2, k1=0..n-j- k2), k2=0..n-j); end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%coefG:6$% \"nG%\"jG6$%#k1G%#k2G6#%)rememberG6\"-%$addG6$-F06$*,)!\"\",&8$\"\"\"8 %F9F9-%*factorialG6#,*9$F99%F6F8F6F:F6F9-%$BinG6$F>F8F9-FB6$,*F?F9F@F6 F8F6F9F9F:F9-FB6$,(F?F9F8F6F:F6F@\"\"#/F8;\"\"!,(F?F9F@F6F:F6/F:;FM,&F ?F9F@F6F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "Coef:=proc(n ,j) local r; option remember; r:=coef(n-2,j);\nif j=0 then r:=r-(-1)^n fi; r; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%CoefG:6$%\"nG%\"jG6 #%\"rG6#%)rememberG6\"C%>8$-%%coefG6$,&9$\"\"\"!\"#F69%@$/F8\"\"!>F0,& F0F6)!\"\"F5F?F0F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 39 " This gives the \+ generating polynomials " }N{XPPEDIT 18 0 "co(n,t):=sum(C(n,j)*t^j,j=0.. n)" ">-%#coG6$%\"nG%\"tG-%$sumG6$*&-%\"CG6$F&%\"jG\"\"\")F'F/F0/F/;\" \"!F&" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "co: =proc(n,t) local j; option remember; add(Coef(n,j)*t^j,j=0..n) end;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#coG:6$%\"nG%\"tG6#%\"jG6#%)remembe rG6\"-%$addG6$*&-%%CoefG6$9$8$\"\"\")9%F6F7/F6;\"\"!F5F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 98 "Here is a short table, again consistent with va lues obtained by exhaustive listing of small cases." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "for nn from 2 to 8 do 'co'(nn,t)=co(nn,t) o d;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#coG6$\"\"#%\"tG\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#coG6$\"\"$%\"tGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#coG6$\"\"%%\"tG,&*$F(\"\"#\"\"\"F(F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#coG6$\"\"&%\"tG,(*$F(\"\"$\"\"\"F(F'*$F( \"\"#\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#coG6$\"\"'%\"tG,,F( \"#?*$F(\"\"#\"#R*$F(\"\"%\"\"\"*$F(\"\"$\"#9F,F0" }}{PARA 1O1 "" 1 "" {XPPMATH 20 "6#/-%#coG6$\"\"(%\"tG,.*$F(\"\"$\"$R\"F(\"$:\"*$F(\"\"&\" \"\"*$F(\"\"%\"#B\"#5F0*$F(\"\"#\"$X#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#coG6$\"\")%\"tG,0*$F(\"\"%\"$`$F(\"$!z*$F(\"\"$\"%S8*$F(\"\"'\" \"\"*$F(\"\"#\"%H=*$F(\"\"&\"#M\"#oF3" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "Results" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 298 9 "Theorem 2" }{TEXT -1 2 ". " }{TEXT 299 11 "The number \+ " }{XPPEDIT 18 0 "C[n,j]" "&%\"CG6$%\"nG%\"jG" }{TEXT -1 1 " " }{TEXT 305 34 "of avoiding paths counted by size " }{XPPEDIT 300 0 "n" "I\"nG 6\"" }{TEXT 301 30 " and by number of outer nodes " }{XPPEDIT 18 0 "j " "I\"jG6\"" }{TEXT 304 15 " satisfies for " }{XPPEDIT 302 0 "j>0" "2 \"\"!%\"jG" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "C_formula;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"CG6$,&%\"nG\"\"\"\"\"#F)%\"jG-%$S umG6$-F-6$*,)!\"\",&%#k1GF)%#k2GF)F)-%*factorialG6#,*F(F)F+F3F5F3F6F3F )-%\"BG6$F:F5F)-F<6$,*F(F)F+F3F5F3F)F)F6F)-F<6$,(F(F)F5F3F6F3F+F*/F5; \"\"!,(F(F)F+F3F6F3/FP6;FF,&F(F)F+F3" }}}{PARA 278 "" 0 "" {TEXT -1 4 " For " }{XPPEDIT 18 0 "j=0" "/%\"jG\"\"!" }{TEXT -1 74 ", the formula s implifies and provides the number of avoiding permutations " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "'Q[n+2]'=subs(j=0,B(n-k1-k2,0)=1,op (2,\"))-(-1)^n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"QG6#,&%\"nG\" \"\"\"\"#F),&-%$SumG6$-F-6$**)!\"\",&%#k1GF)%#k2GF)F)-%*factorialG6#,( F(F)F5F3F6F3F)-%\"BG6$F:F5F)-F<6$,(F(F)F)F)F5F3F6F)/F5;\"\"!,&F(F)F6F3 /F6;FCF(F))F3F(F3" }}}{PARA 0 "" 0 "" {TEXT -1 1 "(" }{XPPEDIT 18 0 "B " "I\"BG6\"" }{TEXT -1 33 " denotes a binomial coefficient.)" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 27 "3. The random graph problem" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 162 "We concl ude this section by showing how to estimate the robustness of connecti ons to link failures between a source and a target in a random graph t hat obeys the " }{XPPEDIT 18 0 "G[n,p]" "&%\"GG6$%\"nG%\"pG" }{TEXT -1 28 " model, where edges between " }{XPPEDIT 18 0 "n" Q"I\"nG6\"" } {TEXT -1 7 " nodes " }{TEXT -1 42 "are chosen independently with proba bility " }{XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "An " }{TEXT 303 13 "avo iding pair" }{TEXT -1 11 " of length " }{XPPEDIT 18 0 "l" "I\"lG6\"" } {TEXT -1 58 " in a graph is an unordered pair of paths, each of length " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 125 " that may share some n odes but are edge disjoint.The mean number of (unordered) avoiding pai rs in a random graph obeying the " }{XPPEDIT 18 0 "G[n,p]" "&%\"GG6$% \"nG%\"pG" }{TEXT -1 155 " model between an arbitrary source and an ar bitary destination is an indicator of the robustness of the graph to a single link failure. This mean value is:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "l:='l': Numpath_formula:=1/2*1/(n*(n-1))*Sum(C(l+1,j) *B(n,l+1+j)*(l+1+j)!,j=0..l)*p^(2*l);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0Numpath_formulaG,$**%\"nG!\"\",&F'\"\"\"F(F*F(-%$SumG6$*(-%\"CG6 $,&%\"lGF*F*F*R%\"jGF*-%\"BG6$F',(F3F*F*F*F4F*F*-%*factorialG6#F8F*/F4; \"\"!F3F*)%\"pG,$F3\"\"#F*#F*FB" }}}{PARA 0 "" 0 "" {TEXT -1 44 "The a rgument is as follows. The coefficient " }{XPPEDIT 18 0 "1/2" "*&\"\" \"\"\"\"\"\"#!\"\"" }{TEXT -1 82 " corresponds to the fact that one ta kes unordered pairs of paths, the coefficient " }{XPPEDIT 18 0 "1/(n*( n-1)" "*&\"\"\"\"\"\"*&%\"nGF$,&F&F$\"\"\"!\"\"F$F)" }{TEXT -1 222 " a verages over all sources and destinations, and the arrangement numbers account for the number of ways to embed an avoiding path into a graph by choosing certain nodes and assigning them in some order to an avoi ding path." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{XPPEDIT 18 0 "C" "I \"CG6\"" }{TEXT -1 47 "-coefficients are as provided by Theorem 2 and \+ " }{XPPEDIT 18 0 "B" "I\"BG6\"" }{TEXT -1 34 " still means binomial co efficient." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The procedure that implements the formula for the mean number o f paths is then" }}{EXCHG {PARA 0 "> " 0 "S" {MPLTEXT 1 0 172 "means:=p roc(n,p,l) local j; option remember; 1/2*1/(n*(n-1))*add(Coef(l+1,j)*b inomial(n,l+1+j)*(l+1+j)!,j=0..l+1)*p^(2*l); expand(\"); if l<=12 then factor(\") else \" fi end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&mean sG:6%%\"nG%\"pG%\"lG6#%\"jG6#%)rememberG6\"C%,$**9$!\"\",&F2\"\"\"F3F5 F3-%$addG6$*(-%%CoefG6$,&9&F5F5F58$F5-%)binomialG6$F2,(F>F5F5F5F?F5F5- %*factorialG6#FCF5/F?;\"\"!F=F5)9%,$F>\"\"#F5#F5FM-%'expandG6#%\"\"G@% 1F>\"#7-%'factorGFQFRF.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "for ll from 1 to 10 do means(n,p,ll) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&%\"nG\" \"\"!\"#F'F',&F&F'!\"$F'F'%\"pG\"\"%#F'\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**,&%\"nG\"\"\"!\"#F'F',&F&F'!\"%F'F',&F&F'!\"$F'\"\" #%\"pG\"\"'#F'F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*.,&%\"nG\"\"\"! \"\"F'F',&F&F'!\"#F'F',&F&F'!\"$F'F',&F&F'!\"%F'F',&F&F'!\"&F'\"\"#%\" pG\"\")#F'F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*.,&%\"nG\"\"\"!\"#F 'F',&F&F'!T\"$F'F',&F&F'!\"%F'F',**$F&\"\"$F'*$F&\"\"#!#6F&\"#D\"#KF'F' ,&F&F'!\"&F'F1%\"pG\"#5#F'F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*0,& %\"nG\"\"\"!\"#F'F',&F&F'!\"$F'F',&F&F'!\"%F'F',&F&F'!\"&F'F',&F&F'!\" 'F'F',.*$F&\"\"&F'*$F&\"\"%!#A*$F&\"\"$\"$i\"*$F&\"\"#!$L$F&!$R*\"%\\L F'F'%\"pG\"#7#F'F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*2,&%\"nG\"\" \"!\"#F'F',&F&F'!\"$F'F',&F&F'!\"%F'F',&F&F'!\"&F'F',&F&F'!\"'F'F',&F& F'!\"(F'F',0*$F&\"\"'F'*$F&\"\"&!#H*$F&\"\"%\"$)H*$F&\"\"$!%H5*$F&\"\" #!%ICF&\"&vO#!&W%RF'F'%\"pG\"#9#F'F@" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#,$*4,&%\"nG\"\"\"!\"#F'F',&F&F'!\"$F'F',&F&F'!\"%F'F',&F&F'!\"&F'F', &F&F'!\"'F'F',&F&F'!\"(F'F',&F&F'!\")F'F',2*$F&\"\"(F'*$F&\"\"'!#P*$F& \"\"&\"$3&*$F&\"\"%!%*p#*$F&\"\"$!%%)H*$F&\"\"#\"&dC*F&!'(RV$\"'()*o$F 'F'%\"pG\"#;#F'FE" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*6,&%\"nG\"\"\" !\"#F'F',&F&F'!\"$F'F',&F&F'!\"%F'F',&F&F'!\"&F'F',&F&F'!\"'F'F',&F&F' !\"(F'F',&F&F'!\")F'F',&F&F'!\"*F'F',4*$F&\"\")F'*$F&\"\"(!#Y*$F&\"\"' \"$:)*$F&\"\"&!%\">'*$F&\"\"%\"%RO*$FU&\"\"$\"'X&f#*$F&\"\"#!(%H\"p\"F& \"(^)pR!(O**R#F'F'%\"pG\"#=#F'FJ" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$ *8,&%\"nG\"\"\"!\"#F'F',&F&F'!\"$F'F',&F&F'!\"%F'F',&F&F'!\"&F'F',&F&F '!\"'F'F',&F&F'!\"(F'F',&F&F'!\")F'F',&F&F'!\"*F'F',&F&F'!#5F'F',6*$F& \"\"*F'*$F&\"\")!#c*$F&\"\"(\"%X7*$F&\"\"'!&$z7*$F&\"\"&\"&&=L*$F&\"\" %\"'vgc*$F&\"\"$!(ks5'*$F&\"\"#\").-=CF&!)$)*pT$!(zuI&F'F'%\"pG\"#?#F' FO" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "He re is an instance for a graph of " }{XPPEDIT 18 0 "n=10^5" "/%\"nG*$ \"#5\"\"&" }{TEXT -1 26 " nodes, a probability of " }{XPPEDIT 18 0 "p =5*10^(-5)" "/%\"pG*&\"\"&\"\"\")\"#5,$\"\"&!\"\"F&" }{TEXT -1 40 ", s o that the mean node degree is about " }{XPPEDIT 18 0 "delta=5" "/%&de ltaG\"\"&" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "seq(subs(n=10^5,p=5.0*10^(-5),[ll,means(n,p,ll)]),ll=2..16);" }} {PARA 12 "" 1 "" {XPPMATH 20 "617$\"\"#$\"+_P%[7$!#<7$\"\"$$\"+TDc6y!# ;7$\"\"%$\"+1Wt_>!#97$\"\"&$\"+RyM\")[!#87$\"\"'$\"+A0>?7!#67$\"\"($\V" +[#\\+0$!#57$\"\")$\"+JJ!Ri(!\"*7$\"\"*$\"+=Fj0>!\"(7$\"#5$\"+7!HJw%! \"'7$\"#6$\"+\"H?0>\"!\"%7$\"#7$\"+ZkevH!\"$7$\"#8$\"+z?.Pu!\"#7$\"#9$ \"+cute=\"\"!7$\"#:$\"+o'\\ak%\"\"\"7$\"#;$\"+03*4;\"F)" }}}{PARA 0 " " 0 "" {TEXT -1 51 "It appears that there is a sharp threshold between " }{XPPEDIT 18 0 "l=7" "/%\"lG\"\"(" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "l=8" "/%\"lG\"\")" }{TEXT -1 101 ". This is to be compared with \+ the number of nodes reachable from the root of a tree with node degree " }{XPPEDIT 18 0 "delta=5" "/%&deltaG\"\"&" }{TEXT -1 5 " in " } {XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 8 " steps: " }{XPPEDIT 18 0 "5^ 7=78125" "/*$\"\"&\"\"(\"&D\"y" }{TEXT -1 15 " is just below " } {XPPEDIT 18 0 "10^5=100000" "/*$\"#5\"\"&\"'++5" }{TEXT -1 7 " while \+ " }{XPPEDIT 18 0 "5^8=390065" "/*$\"\"&\"\")\"'l+R" }{TEXT -1 9 " exce eds " }{XPPEDIT 18 0 "10^5" "*$\"#5\"\"&" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "The probability threshold after wh ich the mean number of unordered paiWrs exceeds 1 is given by" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "p0:=proc(nn,l) local p; fsol ve(subs(n=nn,means(n,p,l))=1,p,0..1) end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p0G:6$%#nnG%\"lG6#%\"pG6\"F+-%'fsolveG6%/-%%subsG6$/ %\"nG9$-%&meansG6%F48$9%\"\"\"F9;\"\"!F;F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 70 "And this quantity is best visualised in terms of the mean node degree:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "delta:=proc (n,p) (n-1)*p; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&deltaG:6$%\" nG%\"pG6\"F)F)*&,&9$\"\"\"!\"\"F-F-9%F-F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 21 "For a graph of size " }{XPPEDIT 18 0 "n=10^5" "/%\"nG*$\"#5\" \"&" }{TEXT -1 9 ", we have" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "seq([ll,delta(10^5,p0(10^5,ll))],ll=2..10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6+7$\"\"#$\"+&\\71w$!\"(7$\"\"$$\"+6%f+@&!\")7$\"\"%$\"+P lDR>F,7$\"\"&$\"+**[zr5F,7$\"\"'$\"+(\\=#=s!\"*7$\"\"($\"+%oIDW&F97$\" \")$\"+l,#QS%F97$\"\"*$\"+9M0NPF97$\"#5$\"+tC%RF$F9" }}}{PARA 0 "" 0 " " {TEXT -1 0 "X" }}{PARA 0 "" 0 "" {TEXT -1 137 "The thresholds are fai rly sharp as attested by plots of the mean values of the number of (un ordered) avoiding pairs when the probability " }{XPPEDIT 18 0 "p" "I\" pG6\"" }{TEXT -1 54 " increases (the horizontal axis gives the mean de gree " }{XPPEDIT 18 0 "delta=(n-1)*p" "/%&deltaG*&,&%\"nG\"\"\"\"\"\"! \"\"F'%\"pGF'" }{TEXT -1 2 ")." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "nn:=10^5: plot([seq(means(nn,delta/(nn-1),ll),ll=3..8)],delta=0. .20,mu=0..1,thickness=3);" }}{PARA 13 "" 1 "" {INLPLOT "6+-%'CURVESG6$ 7V7$\"\"!F(7$$\"39LLLL3VfV!#=$\"3[JtgJd!=V$!#I7$$\"3&pmm;H[D:)F,$\"3\" RHjT@EzY\"!#G7$$\"3LLLLe0$=C\"!#<$\"3d)y7)zllL=!#F7$$\"3jLLL3RBr;F9$\" 3;!z<'\\)\\$*3\"!#E7$$\"3kmm;zjf)4#F9$\"3=$[cg!4)3F%FB7$$\"3TLL$e4;[\\ #F9$\"3/m*)4#3Ab?\"!#D7$$\"3!)****\\i'y]!HF9$\"3*46bp.6&p#Fgn7$$\"3$QLL$3y_qXF9$\"3i!*3zP;mdXFgn7$$\"3]+++ ]1!>+&F9$\"3z'=Csrl)HyFgn7$$\"3J+++]Z/NaYF9$\"3GHHTMAt)G\"!#B7$$\"3<+++ ]$fC&eF9$\"3n[^S+V(*3?F\\p7$$\"3qLL$ez6:B'F9$\"3RuzjM5bFHF\\p7$$\"31nm m;=C#o'F9$\"3@'f3nj*>^WF\\p7$$\"3Nnmmm#pS1(F9$\"3m&[6wa.E@'F\\p7$$\"3< ++]i`A3vF9$\"3f:z%Ri1r&*)F\\p7$$\"3Rmmmm(y8!zF9$\"35I-^$fPm@\"!#A7$$\" 3K,+]i.tK$)F9$\"3/q#y!GEmt;F[r7$$\"3!3++v3zMu)F9$\"3\"o1$pGt#QB#F[r7$$ \"3Zomm\"H_?<*F9$\"3+2A(z!\\vwHF[r7$$\"3,nm;zihl&*F9$\"3w5Cp$*>?IQF[r7 $$\"3:LLL3#G,***F9$\"3$o[3;Rf,(\\F[r7$$\"3WLLezw5V5!#;$\"3`y&[ZS)[SkF[ r7$$\"3.++v$Q#\\\"3\"Fhs$\"3!QaT)4L!***zF[r7$$\"3\\LL$e\"*[H7\"Fhs$\"3 S-L^l^a-5!#@7$$\"3-+++qvxl6Fhs$\"3p5YP!z$)\\D\"Fet7$$\"31++]_qn27Fhs$ \"30^^7jA8^:Fet7$$\"36++Dcp@[7Fhs$\"3L8`F=!p4*=Fet7$$\"3+++]2'HKH\"Fhs $\"3._&Q&z_#)QBFet7$$\"3`mmmwanL8Fhs$\"3zlU*3<([8GFet7$$\"34+++v+'oP\" Fhs$\"3M2*zc:*H1MFet7$$\"3CLLeR<*fT\"Fhs$\"3f==.p)H+.%Fet7$$\"3B+++&)H xe9Fhs$\"3ioh?oz.=[Fet7$$\"3fmm\"H!o-*\\\"Fhs$\"3W;++%RPGn&Fet7$$\"39+ +DTO5T:Fhs$\"3)GZ,$eR%yp'Fet7$$\"3emmmT9C#e\"Fhs$\"3ICN^SZxWyFet7$$\"3 \"****\\i!*3`i\"Fhs$\"3dBdiT,O;#*Fet7$$\"3_LLL$*zym;Fhs$\"3(*o-cjt2Zs5! 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For dense graphs and large size, a numerical pa ttern clearly emerges. Here is what happens for mean degree " } {XPPEDIT 18 0 "delta=10" "/%&deltaG\"#5" }{TEXT -1 10 " and size " } {XPPEDIT 18 0 "n=10^5" "/%\"nG*$\"#5\"\"&" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "seq(subs(n=10^5,p=1.0*10^(-4),[ll,m eans(n,p,ll)]),ll=2..20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "657$\"\"#$ \"+.+v**\\!#;7$\"\"$$\"+E+S**\\!#97$\"\"%$\"+!3+!**\\!#77$\"\"&$\"+(=+ &)*\\!#57$\"\"'b$\"+y.!z*\\!\")7$\"\"($\"+)o+s*\\!\"'7$\"\")$\"+d6S'*\\ !\"%7$\"\"*$\"+K=]&*\\!\"#7$\"#5$\"+nF]%*\\\"\"!7$\"#6$\"+?SS$*\\F$7$ \"#7$\"+bc?#*\\F.7$\"#8$\"+Xx!4*\\F87$\"#9$\"+m.^*)\\FB7$\"#:$\"++O,)) \\FL7$\"#;$\"+PvT')\\FU7$\"#<$\"+qAs%)\\Fgn7$\"#=$\"+,z#H)\\F_o7$\"#>$ \"+NX.\")\\Fgo7$\"#?$\"+'GU!z\\F_p" }}}{PARA 0 "" 0 "" {TEXT -1 83 "Th e mantissas all start with 0.49. This corresponds to a regime where th e distance " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 133 " is small so that only dominant terms in the mean value polynomial intervene. The \+ mean number of avoiding pairs is then approximately" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "mu=1/2*n^(2*l)*p^(2*l+2)*(1+o(1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%#muG,$*()%\"nG,$%\"lG\"\"#\"\"\")%\" pG,&F*F+F+F,F,,&F,F,-%\"oG6#F,F,F,#F,F+" }}}{PARA 0 "" 0 "" {TEXT -1 42 "or, in terms of the mean degree parameter " }{XPPEDIT 18 0 "delta " "I&deltaG6\"" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "mu=1/2*delta^(2*l)*p^2;" }}{PARA 11 "" 1 c"" {XPPMATH 20 "6#/%#mu G,$*&)%&deltaG,$%\"lG\"\"#\"\"\"%\"pGF+#F,F+" }}}{PARA 0 "" 0 "" {TEXT -1 125 "This simplified formula explains the numerical regularit ies seen above, where the mean number increases by a factor of about \+ " }{XPPEDIT 18 0 "100" "\"$+\"" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "l " "I\"lG6\"" }{TEXT -1 16 " increases by 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Conversely, a clear example of \+ " }{TEXT 307 20 "nonasymptotic regime" }{TEXT -1 16 " is provided by \+ " }{XPPEDIT 18 0 "p=10^(1/8)" "/%\"pG)\"#5*&\"\"\"\"\"\"\"\")!\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "n=10^3" "/%\"nG*$\"#5\"\"$" }{TEXT -1 10 ", that is " }{XPPEDIT 18 0 "delta=1.333" "/%&deltaG$\"%L8!\"$" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "nn:=10^3; dd:=10.0^(1/8); pp:=dd/(nn-1);for ll from 10 to 28 by 2 do l=ll,Exact Mean=evalf(subs(n=nn,p=pp,means(n,p,ll)),5),ApproxMean=evalf(1/2*nn^(2 *ll)*pp^(2*ll+2),5) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#nnG\"%+5 " }}{dPARA 11 "" 1 "" {XPPMATH 20 "6#>%#ddG$\"+K9_L8!\"*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#ppG$\"+)Gc[L\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%\"lG\"#5/%*ExactMeanG$\"&VW\"!\")/%+ApproxMeanG$\"&V( GF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%\"lG\"#7/%*ExactMeanG$\"&!yV! \")/%+ApproxMeanG$\"&b7*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%\"lG\" #9/%*ExactMeanG$\"&jJ\"!\"(/%+ApproxMeanG$\"&u*GF*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%/%\"lG\"#;/%*ExactMeanG$\"&c#R!\"(/%+ApproxMeanG$\"&! *>*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%\"lG\"#=/%*ExactMeanG$\"&8; \"!\"'/%+ApproxMeanG$\"&2#HF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%\"l G\"#?/%*ExactMeanG$\"&sS$!\"'/%+ApproxMeanG$\"&IF*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%\"lG\"#A/%*ExactMeanG$\"&b\"**!\"'/%+ApproxMeanG$ \"&U%H!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%\"lG\"#C/%*ExactMeanG$ \"&?'G!\"&/%+ApproxMeanG$\"&vM*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/ %\"lG\"#E/%*ExactMeanG$\"&M>)!\"&/%+ApproxMeanG$\"&y'H!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%\"lG\"#G/%*ExacetMeanG$\"&iK#!\"%/%+ApproxM eanG$\"&DU*F*" }}}{PARA 0 "" 0 "" {TEXT -1 229 "The ratio between the \+ exact formula and the simplified approximation is typically of about 1 to 2 or 1 to 4, with the approximation being systematically optimisti c. An explanation is that, apart from asymptotic approximations in " } {XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 152 ", the simplified formula \+ precisely neglects the exclusion effect on edges and this is permissib le only in dense large graphs having high connectivity. " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Conclusion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Permutations with constrained succes sion gaps are accessible to " }{HYPERLNK 17 "combstruct" 2 "combstruct " "" }{TEXT -1 6 " and " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 507 " . This is achieved via the notion of templates that are nicely d ecomposable in conjunction with the inclusion-exclusion principle that is expressed by a simple integral transformation. Generating functfion s and recurrences result automatically and this leads to explicit bino mial formulae. An application to robustness in a random interconnectio n graph derives from this approach and we have shown how to determine \+ the mean number of edge-disjoint pairs between a source and a target i n such a random graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "31 \+ 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 } eglects the exclusion effect on edges and this is permissib le only in dense large graphs having high connectivity. " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Conclusion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Permutations with constrained succes sion gaps are accessible to " }{HYPERLNK 17 "combstruct" 2 "combstruct " "" }{TEXT -1 6 " and " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 507 " . This is achieved via the notion of templates that are nicely d ecomposable in conjunction with the inclusion-exclusion principle that is expressed by a simple integral transformation. Generating functfdefect?defend#'defin+2(.!>?9D+KQTrVW]bcf/iQiNj!kuyqyוN}]Sx Ej<definablEdefinemathalia<definemathkeyword qSdefinemathsymbol2( definemathtypNjdefinit+ +#'b}Ħx definitenes !kdefintfdeforalg defordiagdefqsumfdefsum 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In particular, the cases of dimensions 2 and 3 are related to problems of phase transitions in the classical I sing model of statistical physics. Solving the general model for self -avoiding walks is an old open problem so that one focusses on approxi mate models. Exactly solvable models exist for various models obtaine d by constraining the admissible walks. More and more is known about \+ self-avoiding walks by relaxing these constraints." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Here we follow a thread started by A. J. Guttmann and T. Prellberg [" }{TEXT 262 83 "Staircase polygons, elliptic integ rals, Heun functions, and lattice Green functions" }{TEXT -1 67 ", (19 93), Physical Review E, (47) 4, R2233-2236]. We show how the " } {HYPERLNK 17 "Gfun" 2 "gfun" "" }{TEXT -1 311 " package may be used to derive in a matter of seconds differential equations that were previo usly guessed in the article above, and how to establish rigorously the singular structure of the O}DE's. As a by-product, we derive effectiv e asymptotic estimates. Early results on this subject date back to G. Polya [" }{TEXT 263 41 "On the number of certain lattice polygons" } {TEXT -1 67 ", (1969), Journal of Combinatorial Theory, Series A, (6), 102-105]." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 184 "A strong but simple constraint on the walks is to require them to proceed by positive steps. Throughou t the remainder of this session, we only consider such paths with posi tive steps." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "A " }{XPPEDIT 278 0 "d" "I\"dG6\"" }{TEXT 279 30 "-dimensional staircase polygon" } {TEXT -1 40 " is a special self-avoiding walk in the " }{XPPEDIT 18 0 "d" "I\"dG6\"" }{TEXT -1 162 "-dimensional integer lattice where all p ositive steps precede all negative steps. This can be interpreted as \+ an (unordered) pair of non-crossing paths of length " }{XPPEDIT 18 0 " n" "I\"nG6\"" }{TEXT -1 69 " from the same origin to the same~ end. He re is an example of length " }{XPPEDIT 18 0 "n=7" "/%\"nG\"\"(" } {TEXT -1 14 " in dimension " }{XPPEDIT 18 0 "d=2" "/%\"dG\"\"#" } {TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "plot([[0,0 ],[2,0],[2,1],[4,1],[4,3],[2,3],[2,2],[1,2],[1,1],[0,1],[0,0]]);" }} {PARA 0 "" 0 "" {INLPLOT "6#-%'CURVESG6$7-7$\"\"!F(7$$\"\"#F(F(7$F*$\" \"\"F(7$$\"\"%F(F-7$F0$\"\"$F(7$F*F37$F*F*7$F-F*7$F-F-7$F(F-F'-%'COLOU RG6&%$RGBG$\"#5!\"\"F(F(" 2 345 212 212 2 0 1 0 2 9 0 4 1 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12010 0 0 0 0 0 0 0 1 1 0 0 0 197 251 0 0 0 0 0 0 }}{PARA 13 "" 1 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "A " }{XPPEDIT 280 0 "d" "I\"dG 6\"" }{TEXT 281 20 "-dimensional festoon" }{TEXT -1 97 " is obtained f rom a sequence of staircase polygons, with the possible introduction o f unit steps." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "Here is an exam ple in dimension 2, involving 3 polygons (thick lines) and 5 unit step s (thin lines)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 729 "\014plo ts[display](\n plot([[0,0],[2,0],[2,2],[1,2],[1,1],[0,1],[0,0]],thi ckness=3),\n plots[textplot]([1/2,3/2,`+`],font=[TIMES,BOLD,18]),\n plots[textplot]([5/2,1/2,`-`],font=[TIMES,BOLD,18]),\n plot([[2 ,2],[3,2]]),\n plot([[3,2],[5,2],[5,5],[6,5],[6,6],[4,6],[4,4],[3,4 ],[3,2]],thickness=3),\n plots[textplot]([7/2,9/2,`+`],font=[TIMES, BOLD,18]),\n plots[textplot]([11/2,7/2,`-`],font=[TIMES,BOLD,18]), \n plot([[6,6],[7,6],[7,7]]),\n plot([[7,7],[10,7],[10,8],[11,8] ,[11,9],[8,9],[8,8],[7,8],[7,7]],thickness=3),\n plots[textplot]([2 1/2,15/2,`+`],font=[TIMES,BOLD,18]),\n plots[textplot]([15/2,17/2,` -`],font=[TIMES,BOLD,18]),\n plot([[11,9],[13,9]]),\nscaling=constr ained,title=`A 2-dimensional festoon`);" }}{PARA 13 "" 1 "" {INLPLOT " 60-%'CURVESG6%7)7$\"\"!F(7$$\"\"#F(F(7$F*F*7$$\"\"\"F(F*7$F.F.7$F(F.F' -%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%*THICKNESSG6#\"\"$-%%TEXTG6%7$$\"+++ ++]!#5$\"+++++:!\"*%\"+G-%%FONTG6%%&TIMESG%%BOLDG\"#=-F>6%7$$\"+++++DF FFA%\"-GFH-F$6$7$F,7$$F6%7$$\"+++++NFF$\"+++++ XFFFGFH-F>6%7$$\"+++++bFFFfoFSFH-F$6$7%F]o7$$\"\"(F(F[o7$FcpFcpF2-F$6% 7+Fep7$$F7F(Fcp7$Fjp$\"\")F(7$$\"#6F(F\\q7$F_q$\"\"*F(7$F\\qFbq7$F\\qF \\q7$FcpF\\qFepF2F9-F>6%7$$\"++++]5!\")$\"+++++vFFFGFH-F>6%7$F]r$\"+++ ++&)FFFSFH-F$6$7$Faq7$$\"#8F(FbqF2-%&TITLEG6#%8A~2-dimensional~festoon G-%(SCALINGG6#%,CONSTRAINEDG" 2 606 606 606 2 0 1 0 2 9 0 4 1 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12010 0 0 0 0 0 0 0 1 1 0 0 0 333 223 0 0 0 0 0 0 }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 247 "We orient each of the staircase polygon (i.e., we distinguish between its two consisting paths), so that a festoon beco mes a sequence of oriented polygons, unit steps remaining non-oriented . On the previous display, this orientation is marked by " }{XPPEDIT 18 0 "`+`" "I\"+G6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "`-`" "I\"-G6 \"" }{TEXT -1 86 " signs.  In this way, we interpret a festoon as an ( oriented) pair of paths of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" } {TEXT -1 26 " with same origin and end." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Denote by " }{XPPEDIT 18 0 "q[d,n]" "&%\"qG6$%\"dG%\"nG" }{TEXT -1 41 " the number of staircase polygons in the " }{XPPEDIT 18 0 "d" "I\"dG6\"" }{TEXT -1 39 "-dimensional integer lattice, and call \+ " }{XPPEDIT 18 0 "Q[d](z)" "-&%\"QG6#%\"dG6#%\"zG" }{TEXT -1 33 " the \+ ordinary generating function" }}}{EXCHG {PARA 260 "" 0 "" {XPPEDIT 18 0 "Q[d](z)=Sum(q[d,n]*z^n,n=0..infinity)" "/-&%\"QG6#%\"dG6#%\"zG-%$Su mG6$*&&%\"qG6$F'%\"nG\"\"\")F)F1F2/F1;\"\"!%)infinityG" }{TEXT -1 1 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "This function has a simple re lation to the ordinary generating function" }}}{EXCHG {PARA 261 "" 0 " " {XPPEDIT 18 0 "P[d](z)=Sum(p[d,n]*z^n,n=0..infinity)" "/-&%\"PG6#%\" dG6#%\"zG-%$SumG6$*&&%\"pG6$F'%\"nG\"\"\")F)F1F2/F1;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "of the number of (ordered) pair s of paths of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 43 " wi th same origin and end. More precisely," }}}{EXCHG {PARA 262 "" 0 "" {XPPEDIT 18 0 "P[d](z)=1/(1+d*z-2*Q[d](z))" "/-&%\"PG6#%\"dG6#%\"zG*& \"\"\"\"\"\",(\"\"\"F,*&F'F,F)F,F,*&\"\"#F,-&%\"QG6#F'6#F)F,!\"\"F7" } {TEXT -1 1 "," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "since any pair o f paths can be viewed as a sequence of pairs of non-crossing paths. E quivalently," }}}{EXCHG {PARA 263 "" 0 "" {XPPEDIT 18 0 "Q[d](z)=(1+d* z)/2-1/(2*P[d](z)" "/-&%\"QG6#%\"dG6#%\"zG,&*&,&\"\"\"\"\"\"*&F'F.F)F. F.F.\"\"#!\"\"F.*&\"\"\"F.*&\"\"#F.-&%\"PG6#F'6#F)F.F1F1" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "In this session, we proceed to compute the singularities and asymptotics of " }{XPPEDIT 18 0 "Q[d ](z)" "-&%\"QG6#%\"dG6#%\"zG" }{TEXT -1 22 " for small dimensions." }} }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "Dimension 2: an explicit case" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 247 "The case of dimension 2 is simp le and yields explicit closed forms for generating functions and count ing sequences, already known to Polya. It is usually solved by hand u sing a Vandermonde convolution. It is readily solved by Maple. The n umber " }{XPPEDIT 18 0 "p[2,n]" "&%\"pG6$\"\"#%\"nG" }{TEXT -1 28 " of pairs of path of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 29 " with same origin and end is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "p[2,n]=Sum(binomial(n,k)^2,k=0..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"pG6$\"\"#%\"nG-%$SumG6$*$-%)binomialG6$F(%\"kGF'/F 0;\"\"!F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "because a path is de termined by the choice of its " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 60 " steps in one of the directions of the lattice. The number " } {XPPEDIT 18 0 "p[2,n]" "&%\"pG6$\"\"#%\"nG" }{TEXT -1 69 " of pairs of paths is thus given by the central binomial coefficient:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "p[2,n]=normal(convert(sum(binomial( n,k)^2,k=0..n),factorial),expanded);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"pG6$\"\"#%\"nG*&-%*factorialG6#,$F(F'\"\"\"-F+6#F(!\"#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "The generating function " } {XPPEDIT 18 0 "P[2](z)" "-&%\"PG6#\"\"#6#%\"zG" }{TEXT -1 32 " of thes e numbers is well-known:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "P[2](z)=sum(op(2,\")*z^n,n=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"PG6#\"\"#6#%\"zG*$,&\"\"\"F-F*!\"%#!\"\"F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Another closed form for the coeffi cients of this series are gotten by Newton's binomial expansion:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "p[2,n]=binomial(-1/2,n)*(-4) ^n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"pG6$\"\"#%\"nG*&-%)binomia lG6$#!\"\"F'F(\"\"\")!\"%F(F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 " The following asymptotics for " }{XPPEDIT 18 0 "p[2,n]" "&%\"pG6$\"\"# %\"nG" }{TEXT -1 22 " is obtained by Maple:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "asympt_p[2]:=expand(simplify(subs(cos(Pi*n)=(-1)^ n,convert(asympt(op(2,\"),n,2),polynom)),symbolic));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)asympt_pG6#\"\"#*(%#PiG#!\"\"F'%\"nGF*)\"\"%F,\" \"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "From the value found for \+ " }{XPPEDIT 18 0 "P[2](z)" "-&%\"PG6#\"\"#6#%\"zG" }{TEXT -1 60 ", we \+ get the generating function for the non-crossing paths:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Q[2](z)=(1+2*z-1/op(2,\"\"\"))/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"QG6#\"\"#6#%\"zG,(#\"\"\"F(F-F *F-*$,&F-F-F*!\"%F,#!\"\"F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "wh ich Maple easily expands into:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "series(op(2,\"),z,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+7% \"zG\"\"#\"\"\"\"\"\"\"\"#F%\"\"$\"\"&\"\"%\"#9\"\"&\"#U\"\"'\"$K\"\" \"(\"$H%\"\")\"%I9\"\"*-%\"OG6#F'\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "A closed form for the coefficients of this series are got ten by Newton's binomial expansion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "q[2,n]=delta[n,0]/2+delta[n,1]-binomial(1/2,n)*(-4)^n /2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"qG6$\"\"#%\"nG,(&%&deltaG6 $F(\"\"!#\"\"\"F'&F+6$F(F/F/*&-%)binomialG6$F.F(F/)!\"%F(F/#!\"\"F'" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "delta[x,y ]" "&%&deltaG6$%\"xG%\"yG" }{TEXT -1 31 " denotes the Kronecker symbol , " }{XPPEDIT 18 0 "delta[x,y]=1" "/&%&deltaG6$%\"xG%\"yG\"\"\"" } {TEXT -1 16 " if and only if " }{XPPEDIT 18 0 "x=y" "/%\"xG%\"yG" } {TEXT -1 69 ", 0 otherwise. The following evaluation corroborates thi s expansion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(-binomi al(1/2,i)*(-4)^i/2,i=2..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*\"\"\" \"\"#\"\"&\"#9\"#U\"$K\"\"$H%\"%I9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The following asymptotics for " }{XPPEDIT 18 0 "q[2,n]" "&%\"qG 6$\"\"#%\"nG" }{TEXT -1 22 " is obtained by Maple:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "asympt_q[2]:=expand(simplify(subs(cos(Pi*n )=(-1)^n,convert(asympt(-(-1)^n*4^n*binomial(1/2,n)/2,n,2),polynom)),s ymbolic));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)asympt_qG6#\"\"#,$*( %#PiG#!\"\"F'%\"nG#!\"$F')\"\"%F-\"\"\"#F2F1" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 40 "We check this asymptotic estimate using " }{HYPERLNK 17 "Gfun" 2 "gfun" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "with(gfun);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7T%(La placeG%.algebraicsubsG%.algeqtodiffeqG%.algeqtoseriesG%.algfuntoalgeqG %&borelG%.cauchyproductG%.diffeq*diffeqG%.diffeq+diffeqG%,diffeqtorecG %)guesseqnG%(guessgfG%0hadamardproductG%0holexprtodiffeqG%)invborelG%, listtoalgeqG%-listtodiffeqG%0listtohypergeomG%+listtolistG%.listtoratp olyG%*listtorecG%-listtoseriesG%5listtoseries/LaplaceG%1listtoseries/e gfG%4listtoseries/lgdegfG%4listtoseries/lgdogfG%1listtoseries/ogfG%4li sttoseries/revegfG%4listtoseries/revogfG%,maxdegcoeffG%*maxdegeqnG%,ma xordereqnG%,mindegcoeffG%*mindegeqnG%,minordereqnG%*optionsgfG%,poltod iffeqG%)poltorecG%/ratpolytocoeffG%(rec*recG%(rec+recG%,rectodiffeqG%* rectoprocG%.seriestoalgeqG%/seriestodiffeqG%2seriestohypergeomG%-serie stolistG%0seriestoratpolyG%,seriestorecG%/seriestoseriesG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "The following first order differential eq uation is satisfied by the generating function " }{XPPEDIT 18 0 "Q[2]( z)=1/2+z-sqrt(1-4*z)/2" "/-&%\"QG6#\"\"#6#%\"zG,(*&\"\"\"\"\"\"\"\"#! \"\"F-F)F-*&-%%sqrtG6#,&\"\"\"F-*&\"\"%F-F)F-F/F-\"\"#F/F/" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "2*f(z)+(1-4*z)*diff (f(z),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%\"fG6#%\"zG\"\"#*&,& \"\"\"F+F'!\"%F+-%%diffG6$F$F'F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "normal(eval(subs(f(z)=sqrt(1-4*z),\")));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "W e compute a recursion on the coefficients " }{XPPEDIT 18 0 "q[2,n]" "& %\"qG6$\"\"#%\"nG" }{TEXT -1 21 " of the series using " }{HYPERLNK 17 "gfun[diffeqtorec]" 2 "gfun[diffeqtorec]" "" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "diffeqtorec(\"\",f(z),u(n)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&\"\"#\"\"\"%\"nG!\"%F'-%\"uG 6#F(F'F'*&,&F(F'F'F'F'-F+6#F.F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Next, " }{HYPERLNK 17 "gfun[rectoproc]" 2 "gfun[rectoproc]" "" } {TEXT -1 72 " returns a procedure that computes the coefficients in an efficient way." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "coeff_of _Q:=rectoproc(\{\",u(0)=1,u(1)=1,u(2)=1\},u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+coeff_of_QG:6#%\"nG6%%\"iG%#u0G%#u1G6\"E\\s$\"\"#\" \"\"\"\"!F/F/F/C%>8%F/?(8$\"\"$F/,&9$F/!\"\"F/%%trueGC$>8&,$*(,&\"\"'F /F5!\"%F/F3F/F5F9F9>F3F=,$*(,&FAF/F8FBF/F3F/F8F9F9F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "We finally check our asymptotic estimate: the following ratio goes to 1 when " }{XPPEDIT 18 0 "n" "I\"nG6\"" } {TEXT -1 18 " goes to infinity." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "for i from 100 to 1000 by 100 do i=evalf(coeff_of_Q(i)/subs(n= i,asympt_q[2]),10) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+\"$\"+M 'pP+\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+#$\"+'*)z=+\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+$$\"+u@D,5!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+%$\"+A(Q4+\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+&$\"+$y]2+\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/\"$+'$\"+Vbi+5!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+($\"+8h` +5!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+)$\"+b!p/+\"!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+*$\"+3pT+5!\"*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/\"%+5$\"+&>v.+\"!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Using " }{HYPERLNK 17 "series" 2 "series" "" }{TEXT -1 11 " to expand " }{XPPEDIT 18 0 "P[2](z)" "-&%\"PG6#\"\"#6#%\"zG" }{TEXT -1 30 " and compute the coefficients " }{XPPEDIT 18 0 "p[2,n]" "&%\"pG 6$\"\"#%\"nG" }{TEXT -1 57 " would be less efficient and require more \+ time and space." }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "In higher di mensions, the problem no longer has a known explicit solution. A natu ral tool to attack it are " }{TEXT 259 27 "Bessel generating functions " }{TEXT -1 48 ". The Bessel generating function of a sequence " } {XPPEDIT 18 0 "s[n]" "&%\"sG6#%\"nG" }{TEXT -1 33 " of numbers is defi ned as the sum" }}}{EXCHG {PARA 264 "" 0 "" {XPPEDIT 18 0 "Sum(s[n]*z^ n/n!^2,n=0..infinity)" "-%$SumG6$*(&%\"sG6#%\"nG\"\"\")%\"zGF)F**$-%*f actorialG6#F)\"\"#!\"\"/F);\"\"!%)infinityG" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "For instance, the Bessel generatin g function of the constant sequence " }{XPPEDIT 18 0 "s[n]=1" "/&%\"sG 6#%\"nG\"\"\"" }{TEXT -1 17 " is given by the " }{HYPERLNK 17 "modifie d Bessel function" 2 "BesselJ" "" }}}{EXCHG {PARA 265 "" 0 "" {XPPEDIT 18 0 "I[0](2*sqrt(z))=Sum(z^n/n!^2,n=0..infinity)" "/-&%\"IG6 #\"\"!6#*&\"\"#\"\"\"-%%sqrtG6#%\"zGF+-%$SumG6$*&)F/%\"nGF+*$-%*factor ialG6#F5\"\"#!\"\"/F5;F'%)infinityG" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "which we denote by j(z). The Bessel generatin g function of the numbers of pairs of paths with same origin and end a nd positive steps in a " }{XPPEDIT 18 0 "d" "I\"dG6\"" }{TEXT -1 44 "- dimensional lattice is given by the product" }}}{EXCHG {PARA 266 "" 0 "" {XPPEDIT 18 0 "Product(j(z[i]),i=1..d)" "-%(ProductG6$-%\"jG6#&%\"z G6#%\"iG/F+;\"\"\"%\"dG" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "z[i]" "&%\"zG6#%\"iG" }{TEXT -1 30 " marks the steps in dimension " }{XPPEDIT 18 0 "i" "I\"iG6\"" } {TEXT -1 65 ". It follows that the Bessel generating function of the \+ numbers " }{XPPEDIT 18 0 "p[d,n]" "&%\"pG6$%\"dG%\"nG" }{TEXT -1 28 " \+ with respect to the length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 17 " of the paths is " }{XPPEDIT 18 0 "j(z)^d" ")-%\"jG6#%\"zG%\"dG" } {TEXT -1 1 "." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 58 "Higher dimensio nal cases: computing differential equations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "In dimension 2, the generating function " }{XPPEDIT 18 0 "P[2](z)" "-&%\"PG6#\"\"#6#%\"zG" }{TEXT -1 38 " is algebraic. In hig her dimensions, " }{XPPEDIT 18 0 "P[d](z)" "-&%\"PG6#%\"dG6#%\"zG" } {TEXT -1 32 " belongs to the larger class of " }{TEXT 260 19 "holonomi c functions" }{TEXT -1 14 ". A function " }{XPPEDIT 18 0 "f(z)" "-%\" fG6#%\"zG" }{TEXT -1 232 " is called holonomic when it satisfies a lin ear differential equation with rational function coefficients. This c lass of functions benefits from numerous closure properties for which \+ algorithms have been implemented in the package " }{HYPERLNK 17 "Gfun " 2 "gfun" "" }{TEXT -1 100 ". In particular, this class is closed un der sum, product, Borel and inverse Borel transforms. The " } {HYPERLNK 17 "Borel transform" 2 "gfun[borel]" "" }{TEXT -1 35 " defin ed in Gfun is related to the " }{HYPERLNK 17 "inverse Laplace transfor m" 2 "inttrans[invlaplace]" "" }{TEXT -1 50 ": formally applied on a f ormal power series, it is" }}}{EXCHG {PARA 267 "" 0 "" {XPPEDIT 18 0 " Borel(Sum(s[n]*z^n,n=0..infinity))=Sum(s[n]*z^n/n!,n=0..infinity)" "/- %&BorelG6#-%$SumG6$*&&%\"sG6#%\"nG\"\"\")%\"zGF-F./F-;\"\"!%)infinityG -F'6$*(&F+6#F-F.)F0F-F.-%*factorialG6#F-!\"\"/F-;F3F4" }{TEXT -1 1 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "This is enough for us to comp ute differential equations satisfied by the " }{XPPEDIT 18 0 "P[d](z) " "-&%\"PG6#%\"dG6#%\"zG" }{TEXT -1 1 "." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Dimension 2 revisited" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "This case was solved above in an explicit way." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "We proceed to get " }{XPPEDIT 18 0 "P[2](z)" "-&% \"PG6#\"\"#6#%\"zG" }{TEXT -1 21 ", and more generally " }{XPPEDIT 18 0 "P[d](z)" "-&%\"PG6#%\"dG6#%\"zG" }{TEXT -1 16 ", by the formula" }} }{EXCHG {PARA 268 "" 0 "" {XPPEDIT 18 0 "P[d](z)=(Borel@@(-2))(j(z)^d) " "/-&%\"PG6#%\"dG6#%\"zG--%#@@G6$%&BorelG,$\"\"#!\"\"6#)-%\"jG6#F)F' " }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "We first comp ute a differential equation satisfied by " }{XPPEDIT 18 0 "j" "I\"jG6 \"" }{TEXT -1 58 " starting from a recurrence satisfied by its coeffic ients " }{XPPEDIT 18 0 "u[n]=1/n!^2" "/&%\"uG6#%\"nG*&\"\"\"\"\"\"*$-% *factorialG6#F&\"\"#!\"\"" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 63 "diff_eq_j:=rectodiffeq(\{(n+1)^2*u(n+1)=u(n),u(0)=1 \},u(n),j(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*diff_eq_jG<$/-%\" jG6#\"\"!\"\"\",(-F(6#%\"zGF+-%%diffG6$F-F/!\"\"*&F/F+-F16$F0F/F+F3" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "We compute equations satisfied b y the powers " }{XPPEDIT 18 0 "j(z)^d" ")-%\"jG6#%\"zG%\"dG" }{TEXT -1 7 " using " }{HYPERLNK 17 "gfun[poltodiffeq]" 2 "gfun[poltodiffeq] " "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "powe r_j:=proc(d) option remember; poltodiffeq(j(z)^d,[diff_eq_j],[j(z)],j( z)) end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "For instance, in the \+ case " }{XPPEDIT 18 0 "d=2" "/%\"dG\"\"#" }{TEXT -1 39 ", the followin g system is satisfied by " }{XPPEDIT 18 0 "j(z)^2" "*$-%\"jG6#%\"zG\" \"#" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "powe r_j(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/-%\"jG6#\"\"!\"\"\"/---% #@@G6$%\"DG\"\"#6#F&F'\"\"$/--F0F2F'F1,*-F&6#%\"zG!\"#*&,&F)F)F:!\"%F) -%%diffG6$F8F:F)F)*&-F@6$-F@6$F?F:F:F)F:F1F)*&F:F)FEF)F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "We now derive a differential system satis fied by " }{XPPEDIT 18 0 "P[d](z)" "-&%\"PG6#%\"dG6#%\"zG" }{TEXT -1 42 ", the ordinary generating function of the " }{XPPEDIT 18 0 "p[d,n] " "&%\"pG6$%\"dG%\"nG" }{TEXT -1 49 ", by computing two inverse Borel \+ transforms with " }{HYPERLNK 17 "gfun[invborel]" 2 "gfun[invborel]" " " }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "double _inv_borel:=proc(sys,f,z) option remember; invborel(invborel(sys,f(z), diffeq),f(z),diffeq) end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "In t he case " }{XPPEDIT 18 0 "d=2" "/%\"dG\"\"#" }{TEXT -1 10 ", we have: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "double_inv_borel(power_ j(2),j,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/-%\"jG6#\"\"!\"\"\",& -F&6#%\"zG\"\"#*&,&!\"\"F)F-\"\"%F)-%%diffG6$F+F-F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "The equation is solve by " }{HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dsolve(\",j(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"jG6#%\"zG*&%\"IG\"\"\",&!\"\"F*F'\"\"%#F,\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "The possible singularities of a holonomic equati on are given by its leading coefficient (on the previous equation, " } {XPPEDIT 18 0 "4*z-1" ",&*&\"\"%\"\"\"%\"zGF%F%\"\"\"!\"\"" }{TEXT -1 75 ") and these singularities yield the asymptotic form of coefficient s by the " }{TEXT 261 30 "method of singularity analysis" }{TEXT -1 55 ". The singularity read on the previous equation is in " } {XPPEDIT 18 0 "1/4" "*&\"\"\"\"\"\"\"\"%!\"\"" }{TEXT -1 39 ", so that the exponential behaviour of " }{XPPEDIT 18 0 "p[2,n]" "&%\"pG6$\"\"# %\"nG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "4^n" ")\"\"%%\"nG" }{TEXT -1 41 ", which we found in the previous section." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 55 "In the case of dimension 2, we read the singularit y by:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "solve(coeff(op(rem ove(type,\"\",equation)),diff(j(z),z)),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"%" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 " Dimension 3" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "We perform the same calculations for dimensions 3 and higher." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "System satisfied by " }{XPPEDIT 18 0 "j(z)^3" "*$-%\"jG6# %\"zG\"\"$" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "power_j(3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<'/-%\"jG6#\"\"!\" \"\",,*&,&!\"$F)%\"zG\"\"*F)-F&6#F.F)F)*&,&F.!#?F)F)F)-%%diffG6$F0F.F) F)*&,&F.\"\"(*$F.\"\"#!#5F)-F66$F5F.F)F)*&-F66$-F66$F>F.F.F)F.\"\"$F)* &FCF)F.F<\"\"'/--%\"DG6#F&F'FE/---%#@@G6$FKF " 0 "" {MPLTEXT 1 0 24 "double_inv_borel(\",j,z);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#<'/-%\"jG6#\"\"!\"\"\",(*&,&%\"zG!\"* \"\"$F)F)-F&6#F-F)F)*&,(*$F-\"\"#!#FF-\"#?!\"\"F)F)-%%diffG6$F0F-F)F)* &,(*$F-F/F.F4\"#5F-F8F)-F:6$F9F-F)F)/---%#@@G6$%\"DGF56#F&F'\"#I/---FF 6$FHF/FIF'\"$e&/--FHFIF'F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Ord er and possible singularities." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "op(sort([solve(coeff(op(remove(type,\",equation)),diff(j(z),z, z)),z)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!#\"\"\"\"\"*F%" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Dimension 4" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 20 "System satisfied by " }{XPPEDIT 18 0 "j(z)^4" "*$- %\"jG6#%\"zG\"\"%" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "power_j(4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<(/-% \"jG6#\"\"!\"\"\"/--%\"DG6#F&F'\"\"%/---%#@@G6$F-F/F.F'#\"$z'\"\"'/--- F46$F-\"\"$F.F'#\"$G\"F>/---F46$F-\"\"#F.F'\"#9,.*&,&%\"zG\"#k!\"%F)F) -F&6#FKF)F)*&,(FK!#o*$FKFFFLF)F)F)-%%diffG6$FNFKF)F)*&,&FS!#!*FK\"#:F) -FU6$FTFKF)F)*&,&*$FKF>!#?FS\"#DF)-FU6$FenFKF)F)*&-FU6$-FU6$F\\oFKFKF) FKF/F)*&FaoF)FKF>\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "System s atisfied by " }{XPPEDIT 18 0 "P[4](z)" "-&%\"PG6#\"\"%6#%\"zG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "double_inv_borel (\",j,z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<(/-%\"jG6#\"\"!\"\"\"/-- -%#@@G6$%\"DG\"\"$6#F&F'\"%O:/---F.6$F0\"\"#F2F'\"#c/---F.6$F0\"\"%F2F '\"&%=l,**&,&%\"zG!#kF@F)F)-F&6#FEF)F)*&,(*$FEF9!$[%FE\"#o!\"\"F)F)-%% diffG6$FGFEF)F)*&,(*$FEF1!$%QFK\"#!*FE!\"$F)-FP6$FOFEF)F)*&,(*$FEF@FFF T\"#?FKFNF)-FP6$FXFEF)F)/--F0F2F'F@" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Order and possible singularities." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "op(sort([solve(coeff(op(remove(type,\",equation)),dif f(j(z),z$3)),z)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"!F##\"\"\" \"#;#F%\"\"%" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Dimension 5" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "System satisfied by " }{XPPEDIT 18 0 "j(z)^5" "*$-%\"jG6#%\"zG\"\"&" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "power_j(5);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<)/-%\"jG6#\"\"!\"\"\"/--%\"DG6#F&F'\"\"&/---%#@@G6$F-F/F.F'#\"% F&)\"\")/---F46$F-\"\"%F.F'#\"%&)y\"#C/---F46$F-\"\"$F.F'#\"$X&\"\"'/- --F46$F-\"\"#F.F'#\"#XFP,0*&,(%\"zG\"$&G!\"&F)*$FVFP!$D#F)-F&6#FVF)F)* &,(FY\"$x(FV!$'>F)F)F)-%%diffG6$FenFVF)F)*&,(FV\"#J*$FVFG\"$f#FY!$=&F) -F\\o6$F[oFVF)F)*&,&FY\"#!*Fao!$!GF)-F\\o6$FdoFVF)F)*&,&Fao\"#l*$FVF>! #NF)-F\\o6$FjoFVF)F)*&-F\\o6$-F\\o6$FapFVFVF)FVF/F)*&FfpF)FVF>\"#:" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "System satisfied by " }{XPPEDIT 18 0 "P[5](z)" "-&%\"PG6#\"\"&6#%\"zG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "double_inv_borel(\",j,z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<)/-%\"jG6#\"\"!\"\"\"/---%#@@G6$%\"DG\"\"%6 #F&F'\"'S#*=/--F0F2F'\"\"&/---F.6$F0\"\"$F2F'\"%qK/---F.6$F0\"\"#F2F' \"#!*/---F.6$F0F7F2F'\")+'[`\",,*&,(*$%\"zGFD\"$+*FP!$&GF7F)F)-F&6#FPF )F)*&,**$FPF=\"%+sFO!%jRFP\"$'>!\"\"F)F)-%%diffG6$FSFPF)F)*&,**$FPF1\" %]&)FW!%,lFO\"$=&FP!\"(F)-Fgn6$FfnFPF)F)*&,**$FPF7\"%+FF[o!%!f#FW\"$!G FO!\"'F)-Fgn6$F`oFPF)F)*&,**$FP\"\"'\"$D#Fdo!$f#F[o\"#NFWFenF)-Fgn6$Fi oFPF)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Order and possible sin gularities." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "op(sort([sol ve(coeff(op(remove(type,\",equation)),diff(j(z),z$4)),z)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(\"\"!F#F##\"\"\"\"#D#F%\"\"*F%" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Dimension 6" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "System satisfied by " }{XPPEDIT 18 0 "j(z)^6" "*$-%\"jG6# %\"zG\"\"'" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "power_j(6);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<*/-%\"jG6#\"\"!\" \"\"/--%\"DG6#F&F'\"\"'/---%#@@G6$F-\"\"$F.F'\"$m\"/---F46$F-\"\"#F.F' \"#L/---F46$F-F/F.F'#\"'`tt\"#g/---F46$F-\"\"%F.F'#\"%^IFL/---F46$F-\" \"&F.F'#\"&8?$\"#5,2*&,(%\"zG\"%?5!\"'F)*$FenF=!%cMF)-F&6#FenF)F)*&,*F en!$;&Fhn\"%CdF)F)*$FenF6!%/BF)-%%diffG6$FjnFenF)F)*&,(F`o\"%/ZFhn!%OC Fen\"#jF)-Fco6$FboFenF)F)*&,(Fhn\"$,$*$FenFL\"$%yF`o!%3CF)-Fco6$FjoFen F)F)*&,&F`o\"$]$F_p!$+(F)-Fco6$FbpFenF)F)*&,&*$FenFT!#cF_p\"$S\"F)-Fco 6$FhpFenF)F)*&-Fco6$-Fco6$F_qFenFenF)FenF/F)*&FdqF)FenFT\"#@" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "System satisfied by " }{XPPEDIT 18 0 "P[6](z)" "-&%\"PG6#\"\"'6#%\"zG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "double_inv_borel(\",j,z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<*/-%\"jG6#\"\"!\"\"\"/---%#@@G6$%\"DG\"\"#6 #F&F'\"$K\"/---F.6$F0\"\"$F2F'\"%wf/---F.6$F0\"\"%F2F'\"'W$R%/---F.6$F 0\"\"&F2F'\")?()4Y,.*&,(*$%\"zGF1\"&CQ\"FM!%?5\"\"'F)F)-F&6#FMF)F)*&,* *$FMF9\"'ON>FL!&cf#FM\"$;&!\"\"F)F)-%%diffG6$FQFMF)F)*&,**$FMF@\"'g,QF U!&w>(FL\"%OCFM!#:F)-Fen6$FZFMF)F)*&,*FL!#DFU\"%3C*$FMFG\"'o>@Fin!&'>^ F)-Fen6$F^oFMF)F)*&,*FU!#5Fin\"$+(*$FMFP\"&?.%Fdo!&g<\"F)-Fen6$FgoFMF) F)*&,*FinFY*$FM\"\"(\"%/BF]p!$%yFdo\"#cF)-Fen6$F`pFMF)F)/--F0F2F'FP/-- -F.6$F0FPF2F'\"+?*H2P'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Order a nd possible singularities." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "op(sort([solve(coeff(op(remove(type,\",equation)),diff(j(z),z$5)), z)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)\"\"!F#F#F##\"\"\"\"#O#F%\"# ;#F%\"\"%" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Dimension 7" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "System satisfied by " }{XPPEDIT 18 0 "j(z)^7." ")-%\"jG6#%\"zG$\"\"(\"\"!" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "power_j(7);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<+/- %\"jG6#\"\"!\"\"\"/---%#@@G6$%\"DG\"\"%6#F&F'#\"&:n$\"#C/---F.6$F0\"\" $F2F'#\"%X;\"\"'/---F.6$F0\"\"#F2F'#\"#\"*FD/--F0F2F'\"\"(/---F.6$F0FJ F2F'#\"*,-n?\"\"$?(/---F.6$F0F>F2F'#\")p,@FFR/---F.6$F0\"\"&F2F'#\"'d \"[*\"$?\",4*&,**$%\"zGF;\"&D5\"!\"(F)Fao\"%8K*$FaoFD!&55$F)-F&6#FaoF) F)*&,*Fao!%%G\"F`o!&k;&Feo\"&uJ$F)F)F)-%%diffG6$FgoFaoF)F)*&,*Fao\"$F \"F`o\"&Q0&*$FaoF1!&;H\"Feo!&=-\"F)-F_p6$F^pFaoF)F)*&,(F`o!&kk\"Feo\"$ m*Fep\"&S(>F)-F_p6$FhpFaoF)F)*&,(Fep!%e$)F`o\"%,<*$FaoFin\"%u>F)-F_p6$ F_qFaoF)F)*&,&Fep\"%]5Feq!%7:F)-F_p6$FgqFaoF)F)*&,&Feq\"$m#*$FaoF>!#%) F)-F_p6$F]rFaoF)F)*&-F_p6$-F_p6$FdrFaoFaoF)FaoFJF)*&FirF)FaoF>\"#G" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "System satisfied by " }{XPPEDIT 18 0 "P[7](z)" "-&%\"PG6#\"\"(6#%\"zG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "double_inv_borel(\",j,z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<+/---%#@@G6$%\"DG\"\"'6#%\"jG6#\"\"!\",!o@8 f>/-F-F.\"\"\",0*&,**$%\"zG\"\"$!'+Qz*$F8\"\"#\"'!3[#F8!%Ek\"#9F3F3-F- 6#F8F3F3*&,,*$F8\"\"%!)+q!>\"F7\"([Wq&F;!'qYGF8\"%oD!\"#F3F3-%%diffG6$ F@F8F3F3*&,,*$F8\"\"&!)+hQFFD\")#*fBFP\")/N+:FD!(gbS\"F7\"&GH$F;!$!=F3-FL6$FWF8F3F3* &,,*$F8\"\"(!(]\"e`Fen\"(sn'[FP!''4b&FD\"&;n\"F7!$I\"F3-FL6$F[oF8F3F3* &,,*$F8\"\")!']`fF_o\"'o*>'Fen!&3H)FP\"%CIFD!#IF3-FL6$FfoF8F3F3*&,,*$F 8\"\"*!&]?#Fjo\"&Ke#F_o!%[RFen\"$o\"FPFJF3-FL6$FapF8F3F3/---F(6$F*FQF, F.\"*S)yP6/---F(6$F*FEF,F.\"'g6))/---F(6$F*F9F,F.\"%q)*/---F(6$F*F " 0 "" {MPLTEXT 1 0 71 "op(sort([solve(coeff(op(remove(type,\",eq uation)),diff(j(z),z$6)),z)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+\" \"!F#F#F#F##\"\"\"\"#\\#F%\"#D#F%\"\"*F%" }}}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "These experimental results suggest a pattern for the sing ularities of the equation in dimension " }{XPPEDIT 18 0 "d" "I\"dG6\" " }{TEXT -1 86 ". More specifically, the leading coefficient of the d ifferential equation is given by" }}}{EXCHG {PARA 272 "" 0 "" {XPPEDIT 18 0 "z^(d-2)*Product(1-(2*k+1)*z,k=0..p)" "*&)%\"zG,&%\"dG\" \"\"\"\"#!\"\"F'-%(ProductG6$,&\"\"\"F'*&,&*&\"\"#F'%\"kGF'F'\"\"\"F'F 'F$F'F)/F3;\"\"!%\"pGF'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "when " }{XPPEDIT 18 0 "d=2*p+1" "/%\"dG,&*&\"\"#\"\"\"%\"pGF'F'\"\"\"F'" } {TEXT -1 8 ", and by" }}}{EXCHG {PARA 273 "" 0 "" {XPPEDIT 18 0 "z^(d- 2)*Product(1-2*k*z,k=1..p)" "*&)%\"zG,&%\"dG\"\"\"\"\"#!\"\"F'-%(Produ ctG6$,&\"\"\"F'*(\"\"#F'%\"kGF'F$F'F)/F1;\"\"\"%\"pGF'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "when " }{XPPEDIT 18 0 "d=2*p" "/%\"dG*&\" \"#\"\"\"%\"pGF&" }{TEXT -1 1 "." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 33 "The singular structure of the ODE" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The equation used to describe " }{XPPEDIT 18 0 "j(z)" "-%\"jG6# %\"zG" }{TEXT -1 36 " introduces a \"parasitic\" function, " } {XPPEDIT 18 0 "k(z)=K[0](2*sqrt(z))" "/-%\"kG6#%\"zG-&%\"KG6#\"\"!6#*& \"\"#\"\"\"-%%sqrtG6#F&F/" }{TEXT -1 56 ", also expressed in terms of \+ a modified Bessel function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "op(remove(type,diff_eq_j,equation));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%\"jG6#%\"zG\"\"\"-%%diffG6$F$F'!\"\"*&F'F(-F*6$F)F'F(F," }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Both " }{XPPEDIT 18 0 "j(z)" "-%\"j G6#%\"zG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "k(z)" "-%\"kG6#%\"zG" } {TEXT -1 30 " are solutions found my Maple:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dsolve(\",j(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%\"jG6#%\"zG,&*&%$_C1G\"\"\"-%(BesselIG6$\"\"!,$*$F'#F+\"\"#F3F+F+ *&%$_C2GF+-%(BesselKGF.F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Th e equation above is satisfied by " }{TEXT 256 3 "any" }{TEXT -1 23 " l inear combination of " }{XPPEDIT 18 0 "j(z)" "-%\"jG6#%\"zG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "k(z)" "-%\"kG6#%\"zG" }{TEXT -1 51 ". In the same way, the equation used to described " }{XPPEDIT 18 0 "j(z)^d " ")-%\"jG6#%\"zG%\"dG" }{TEXT -1 17 " is satisfied by " }{TEXT 257 3 "any" }{TEXT -1 36 " linear combination of the products " }{XPPEDIT 18 0 "j(z)^l*k(z)^(d-l)" "*&)-%\"jG6#%\"zG%\"lG\"\"\")-%\"kG6#F',&%\"d GF)F(!\"\"F)" }{TEXT -1 54 ". For instance, this is easily checked fo r the cases " }{XPPEDIT 18 0 "d=2" "/%\"dG\"\"#" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "d=3" "/%\"dG\"\"$" }{TEXT -1 1 "." }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 11 "Dimension 2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Here is the differential equation satisfied by " }{XPPEDIT 18 0 "j(z) ^2" "*$-%\"jG6#%\"zG\"\"#" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "op(remove(type,power_j(2),equation));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,*-%\"jG6#%\"zG!\"#*&,&\"\"\"F+F'!\"%F+-%%diffG6 $F$F'F+F+*&-F.6$-F.6$F-F'F'F+F'\"\"#F+*&F'F+F3F+\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Maple is no longer able to solve in terms of special functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "d solve(\",j(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"jG6#%\"zG-%&DE SolG6$<#,**&F'\"\"#-%%diffG6$-F06$-F06$-%#_YGF&F'F'F'\"\"\"F8*&F'F8F2F 8\"\"$*&,&F8F8F'!\"%F8F4F8F8F6!\"#<#F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "(The development version of Maple is able to solve.)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "We successively plug " }{XPPEDIT 18 0 "j(z)^2" "*$-%\"jG6#%\"zG\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 " k(z)^2" "*$-%\"kG6#%\"zG\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "j(z )*k(z)" "*&-%\"jG6#%\"zG\"\"\"-%\"kG6#F&F'" }{TEXT -1 28 " into the pr evious equation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "op(remo ve(type,power_j(2),equation)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "normal(eval(subs(j=unapply(BesselI(0,2*sqrt(z))^2,z),\")));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "normal(eval(subs(j=unapply(BesselK(0,2*sqrt(z))^2,z), \"\")));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 79 "normal(eval(subs(j=unapply(BesselI(0,2*sqrt( z))*BesselK(0,2*sqrt(z)),z),\"\"\")));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Each evaluation yi elds 0, proving that the function is a solution." }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Dimension 3" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Here is the differential equation satisfied by " }{XPPEDIT 18 0 "j (z)^3" "*$-%\"jG6#%\"zG\"\"$" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "op(remove(type,power_j(3),equation));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&,&!\"$\"\"\"%\"zG\"\"*F'-%\"jG6#F(F'F'*& ,&F(!#?F'F'F'-%%diffG6$F*F(F'F'*&,&F(\"\"(*$F(\"\"#!#5F'-F16$F0F(F'F'* &-F16$-F16$F9F(F(F'F(\"\"$F'*&F>F'F(F7\"\"'" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 21 "We successively plug " }{XPPEDIT 18 0 "j(z)^3" "*$-%\"j G6#%\"zG\"\"$" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "j(z)^2*k(z)" "*&-%\"jG 6#%\"zG\"\"#-%\"kG6#F&\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "j(z)*k( z)^2" "*&-%\"jG6#%\"zG\"\"\"*$-%\"kG6#F&\"\"#F'" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "k(z)^3" "*$-%\"kG6#%\"zG\"\"$" }{TEXT -1 28 " into the \+ previous equation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "norma l(eval(subs(j=unapply(BesselI(0,2*sqrt(z))^3,z),\")));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "normal(eval(subs(j=unapply(BesselI(0,2*sqrt(z))^2*BesselK(0,2*sqrt (z)),z),\"\")));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "\"\"\":" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 79 "normal(eval(subs(j=unapply(BesselI(0,2*sqrt(z))*Bes selK(0,2*sqrt(z))^2,z),\")));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "normal(eval(subs(j=unap ply(BesselK(0,2*sqrt(z))^3,z),\"\")));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "The ordinary gene rating function " }{XPPEDIT 18 0 "P[d](z)" "-&%\"PG6#%\"dG6#%\"zG" } {TEXT -1 42 " is the double inverse Borel transform of " }{XPPEDIT 18 0 "j(z)^d" ")-%\"jG6#%\"zG%\"dG" }{TEXT -1 72 ". In the same way as a bove, the differential equation that vanishes at " }{XPPEDIT 18 0 "P[d ](z)" "-&%\"PG6#%\"dG6#%\"zG" }{TEXT -1 62 " also vanishes at each of \+ the double inverse Borel transforms " }{XPPEDIT 18 0 "(Borel@@(-2))(j( z)^l*k(z)^(d-l))" "--%#@@G6$%&BorelG,$\"\"#!\"\"6#*&)-%\"jG6#%\"zG%\"l G\"\"\")-%\"kG6#F0,&%\"dGF2F1F)F2" }{TEXT -1 62 ". Those transform ha ve a nice integral representation, namely" }}}{EXCHG {PARA 269 "" 0 " " {XPPEDIT 18 0 "(Borel@@(-2))(j(z)^l*k(z)^(d-l))=2*Int(j(z*t)^l*k(z*t )^(d-l)*k(t),t=0...infinity)" "/--%#@@G6$%&BorelG,$\"\"#!\"\"6#*&)-%\" jG6#%\"zG%\"lG\"\"\")-%\"kG6#F1,&%\"dGF3F2F*F3*&\"\"#F3-%$IntG6$*()-F/ 6#*&F1F3%\"tGF3F2F3)-F66#*&F1F3FDF3,&F9F3F2F*F3-F66#FDF3/FD;\"\"!%)inf inityGF3" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "This integral is always convergent at its lower bound. At infinity howeve r, the integrand behaves like" }}}{EXCHG {PARA 270 "" 0 "" {XPPEDIT 18 0 "exp(2*((2*l-d)*sqrt(z)-1)*sqrt(t))/t^((d+1)/4)" "*&-%$expG6#*(\" \"#\"\"\",&*&,&*&\"\"#F(%\"lGF(F(%\"dG!\"\"F(-%%sqrtG6#%\"zGF(F(\"\"\" F0F(-F26#%\"tGF(F()F8*&,&F/F(\"\"\"F(F(\"\"%F0F0" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "It follows that either " } {XPPEDIT 18 0 "2l<=d" "1*&\"\"#\"\"\"%\"lGF%%\"dG" }{TEXT -1 50 " and \+ the integral is defined for any non-negative " }{XPPEDIT 18 0 "z" "I\" zG6\"" }{TEXT -1 5 ", or " }{XPPEDIT 18 0 "d<2*l" "2%\"dG*&\"\"#\"\"\" %\"lGF&" }{TEXT -1 33 " and the integral is defined for " }{XPPEDIT 18 0 "z" "I\"zG6\"" }{TEXT -1 15 " between 0 and " }{XPPEDIT 18 0 "(d- 2*l)^(-2)" "),&%\"dG\"\"\"*&\"\"#F%%\"lGF%!\"\",$\"\"#F)" }{TEXT -1 63 ". We thus obtain all positive singularities of the conjecture." } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 8 "Theorem:" }{TEXT -1 66 " Any homogeneous linear ODE sati sfied by the generating function " }{XPPEDIT 265 0 "P[d](z)" "-&%\"PG6 #%\"dG6#%\"zG" }{TEXT 267 20 " of festoons on the " }{XPPEDIT 268 0 "d " "I\"dG6\"" }{TEXT 269 49 "-dimensional integer lattice is singular a t each " }{XPPEDIT 270 0 "(d-2*l)^(-2)" "),&%\"dG\"\"\"*&\"\"#F%%\"lGF %!\"\",$\"\"#F)" }{TEXT 271 6 ", for " }{XPPEDIT 272 0 "d<2*l" "2%\"dG *&\"\"#\"\"\"%\"lGF&" }{TEXT -1 35 ". More specifically, the function " }{XPPEDIT 266 0 "P[d](z)" "-&%\"PG6#%\"dG6#%\"zG" }{TEXT 273 16 " i s singular at " }{XPPEDIT 274 0 "d^(-2)" ")%\"dG,$\"\"#!\"\"" }{TEXT 275 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 47 "Computing the asymptotics: a connection problem" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "P [d](z)" "-&%\"PG6#%\"dG6#%\"zG" }{TEXT -1 269 " is analytic in a neig hbourhoud of 0, where it is given by its Taylor expansion. On the oth er hand, its dominant singularity (i.e., its singularity closest to 0) and its asymptotic behaviour there are a priori not known. It is nat ural to estimate the asymptotics of " }{XPPEDIT 18 0 "P[d](z)" "-&%\"P G6#%\"dG6#%\"zG" }{TEXT -1 39 " at the smallest positive singularity, \+ " }{XPPEDIT 18 0 "s=d^(-2)" "/%\"sG)%\"dG,$\"\"#!\"\"" }{TEXT -1 60 ", of the equation defining this function. To do so, we use " }{TEXT 283 20 "singularity analysis" }{TEXT -1 3 ". " }{HYPERLNK 17 "First" 1 "" "evaluating the Taylor series" }{TEXT -1 39 ", we evaluate the Ta ylor series at 0. " }{HYPERLNK 17 "Next" 1 "" "computing a basis" } {TEXT -1 78 ", we algorithmically compute a basis of all possible asym ptotic behaviours at " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 32 " of solutions of the equation. " }{HYPERLNK 17 "Finally" 1 "" "the conne ction" }{TEXT -1 105 ", we numerically connect the Taylor series at 0 \+ and a generic linear combination of these asymptotics at " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 50 ", i.e., we equate values at a point \+ between 0 and " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "A common case of singularity of ho lonomic function is that of a " }{TEXT 276 22 "regular singular point " }{TEXT -1 134 ". This is a singular point where the function has a p olynomial behaviour: at such points, the theory expects an expansion o f the form " }{XPPEDIT 18 0 "kappa*z^(p/r)*phi(z^(1/r))" "*(%&kappaG\" \"\")%\"zG*&%\"pGF$%\"rG!\"\"F$-%$phiG6#)F&*&\"\"\"F$F)F*F$" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "phi" "I$phiG6\"" }{TEXT -1 149 " is a \+ formal power series. It is therefore natural to look for such an asym ptotics by a method of \"undetermined exponent\". The possible values for " }{XPPEDIT 18 0 "eta=p/r" "/%$etaG*&%\"pG\"\"\"%\"rG!\"\"" } {TEXT -1 32 " are given by a polynomial, the " }{TEXT 277 19 "indicial polynomial" }{TEXT -1 9 ": a zero " }{XPPEDIT 18 0 "eta" "I$etaG6\"" }{TEXT -1 17 " of multiplicity " }{XPPEDIT 18 0 "m+1" ",&%\"mG\"\"\"\" \"\"F$" }{TEXT -1 68 " of the indicial polynomial indicates solutions \+ with the behaviours " }{XPPEDIT 18 0 "z^eta" ")%\"zG%$etaG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "z^eta*ln(z)" "*&)%\"zG%$etaG\"\"\"-%#lnG6#F$F& " }{TEXT -1 11 ", ..., and " }{XPPEDIT 18 0 "z^eta*ln(z)^m" "*&)%\"zG% $etaG\"\"\")-%#lnG6#F$%\"mGF&" }{TEXT -1 88 ". The following procedu re computes the indicial polynomial of a differential equation." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 247 "indicial_poly:=proc(sys,f,z ) local zero; global eta; option remember;\n zero:=collect(normal(e val(subs(f=proc(z) global eta; z^eta end,op(remove(type,sys,equation)) )))/z^eta,z,normal);\n factor(primpart(coeff(zero,z,ldegree(zero,z) ),eta))\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "As a first remar k, note that " }{XPPEDIT 18 0 "P[d](z)" "-&%\"PG6#%\"dG6#%\"zG" } {TEXT -1 66 " is the only solution of the equation analytic at 0 and s uch that " }{XPPEDIT 18 0 "P[d](0)=1" "/-&%\"PG6#%\"dG6#\"\"!\"\"\"" } {TEXT -1 89 ". To obtain this, let us compute the indicial polynomial s at 0 for the first dimensions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "for i from 2 to 7 do i=indicial_poly(double_inv_borel (power_j(i),j,z),j,z) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"#,$% $etaG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"$,$*$%$etaG\"\"#!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"%,$*$%$etaG\"\"$!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"&,$*$%$etaG\"\"%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"',$*$%$etaG\"\"&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"(,$*$%$etaG\"\"'!\"\"" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 37 "Therefore, the equation satisfied by " }{XPPEDIT 18 0 " P[d](z)" "-&%\"PG6#%\"dG6#%\"zG" }{TEXT -1 14 " in dimension " } {XPPEDIT 18 0 "d" "I\"dG6\"" }{TEXT -1 45 " has one solution which is \+ analytic at 0 and " }{XPPEDIT 18 0 "d-1" ",&%\"dG\"\"\"\"\"\"!\"\"" } {TEXT -1 56 " that are singular at 0, with logarithmic singularities. " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "Connection and asymptotics i n dimension 3" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The smallest sing ularity is " }{XPPEDIT 18 0 "s=1/9" "/%\"sG*&\"\"\"\"\"\"\"\"*!\"\"" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "op(sort([ solve(coeff(op(remove(type,double_inv_borel(power_j(3),j,z),equation)) ,diff(j(z),z,z)),z)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!#\"\" \"\"\"*F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "We first expand the \+ series at 0 to evaluate " }{XPPEDIT 18 0 "P[3](z)" "-&%\"PG6#\"\"$6#% \"zG" }{TEXT -1 10 " at, say, " }{XPPEDIT 18 0 "1/18" "*&\"\"\"\"\"\" \"#=!\"\"" }{TEXT -1 41 ". Next, we recenter the equation around " } {XPPEDIT 18 0 "1/9" "*&\"\"\"\"\"\"\"\"*!\"\"" }{TEXT -1 69 " to study the local behaviour of the solutions there more explicitly." }}} {SECT 1 {PARA 5 "" 0 "evaluating the Taylor series" {TEXT -1 28 "Evalu ating the Taylor series" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 248 "To eva luate the Taylor series, we compute a truncation of it. We actually n eed truncations to several orders to illustrate the slow convergence o f the series close to its singularity. We base on the following ident ity to compute such truncations:" }}}{EXCHG {PARA 271 "" 0 "" {XPPEDIT 18 0 "[z^n]*(phi(t*z)/(1-z))=Sum(c[k]*t^k,k=0..n)" "/*&7#)%\" zG%\"nG\"\"\"*&-%$phiG6#*&%\"tGF(F&F(F(,&\"\"\"F(F&!\"\"F1F(-%$SumG6$* &&%\"cG6#%\"kGF()F.F9F(/F9;\"\"!F'" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "[z^n]*f(z)" "*&7#)%\"zG%\" nG\"\"\"-%\"fG6#F%F'" }{TEXT -1 28 " denotes the coefficient of " } {XPPEDIT 18 0 "z^n" ")%\"zG%\"nG" }{TEXT -1 15 " in the series " } {XPPEDIT 18 0 "f(z)" "-%\"fG6#%\"zG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi" "I$phiG6\"" }{TEXT -1 15 " is any series " }{XPPEDIT 18 0 "phi (z)=Sum(c[n]*z^n,n=0..infinity)" "/-%$phiG6#%\"zG-%$SumG6$*&&%\"cG6#% \"nG\"\"\")F&F.F//F.;\"\"!%)infinityG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "We start from the equation for " } {XPPEDIT 18 0 "P[3](z)" "-&%\"PG6#\"\"$6#%\"zG" }{TEXT -1 1 ":" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "double_inv_borel(power_j(3), j,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<'/-%\"jG6#\"\"!\"\"\",(*&,&% \"zG!\"*\"\"$F)F)-F&6#F-F)F)*&,(*$F-\"\"#!#FF-\"#?!\"\"F)F)-%%diffG6$F 0F-F)F)*&,(*$F-F/F.F4\"#5F-F8F)-F:6$F9F-F)F)/---%#@@G6$%\"DGF56#F&F'\" #I/---FF6$FHF/FIF'\"$e&/--FHFIF'F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "We compute " }{XPPEDIT 18 0 "P[3](t*z)" "-&%\"PG6#\"\"$6#*&%\"t G\"\"\"%\"zGF*" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "algebraicsubs(\",j=z*t,j(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/-%\"jG6#\"\"!\"\"\",(*&,&%\"tG!\"$*&F-\"\"#%\"zGF)\"\"*F)-F&6#F1 F)F)*&,(F)F)*&F1F)F-F)!#?*&F1F0F-F0\"#FF)-%%diffG6$F3F1F)F)*&,(F1F)*&F 1F0F-F)!#5*&F1\"\"$F-F0F2F)-F<6$F;F1F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "We divide by " }{XPPEDIT 18 0 "(1-z)" ",&\"\"\"\"\"\"%\"z G!\"\"" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "p oltodiffeq(j(z)/(1-z),[\"],[j(z)],j(z));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<%/-%\"jG6#\"\"!\"\"\"/--%\"DG6#F&F',&%\"tG\"\"$F)F),(*&,,*&%\"z G\"\"#F0F7\"#O*&F0F7F6F)!\"**&F6F)F0F)!#BF)F)F0F1F)-F&6#F6F)F)*&,.*&F6 F1F0F7\"#XF5!#F*&F6F7F0F)!#SF;\"#?F6F1!\"\"F)F)-%%diffG6$F=F6F)F)*&,.* $F6F7F)*&F6F1F0F)!#5*&F6\"\"%F0F7\"\"*F6FGFD\"#5FAF:F)-FI6$FHF6F)F)" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "We extract its coefficients:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "rec_j:=diffeqtorec(\",j(z), u(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&rec_jG<&/-%\"uG6#\"\"#,(* $%\"tGF*\"#:F-\"\"$\"\"\"F0,**&,(F,\"#O*&F-F*%\"nGF0F4*&F-F*F6F*\"\"*F 0-F(6#F6F0F0*&,**&,&F-!#5F,!\"*F0F6F*F0*&,&F-!#]F,!#OF0F6F0F0F,FDF-!#j F0-F(6#,&F6F0F0F0F0F0*&,**&,&F0F0F-\"#5F0F6F*F0*&,&\"\"'F0F-\"#]F0F6F0 F0F-\"#jF8F0F0-F(6#,&F6F0F*F0F0F0*&,(F6!\"'F@F0*$F6F*!\"\"F0-F(6#,&F6F 0F/F0F0F0/-F(6#F0,&F-F/F0F0/-F(6#\"\"!F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "And we compute a procedure, which to " }{XPPEDIT 18 0 "`` (n,t)" "-%!G6$%\"nG%\"tG" }{TEXT -1 36 " associates the truncated seri es at " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 14 " to the order " } {XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "series_j:=rectoproc(rec_j,u(n),params=[t]):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Next, the series can be computed f ormally at " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "normal(series_j(4,t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$%\"tG\"\"$\"#$**$F%\"\"#\"#:*$F%\"\"%\"$R'F %F&\"\"\"F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "or numerically at \+ a rational number:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "serie s_j(4,1/16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"&\"z$)\"&Ob'" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "(This is more efficient than compu ting the series, and next evaluating it numerically.)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 143 "The following plots display the Taylor e xpansion for increasing an number of terms. Note the behaviour of the series close to the singularity " }{XPPEDIT 18 0 "1/9" "*&\"\"\"\"\" \"\"\"*!\"\"" }{TEXT -1 38 " when we increase the number of terms." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 300 "pl[Taylor]:=plots[display] (\n plot('series_j'(10,z),z=0..1/9,y=0..5,style=point,color=red),\n plot('series_j'(20,z),z=0..1/9,y=0..5,style=point,color=green),\n \+ plot('series_j'(100,z),z=0..1/9,y=0..5,style=point,color=blue),\nti tle=`Taylor series (red=10 terms, green=20 terms, blue=100 terms)`):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "pl[Taylor];" }}{PARA 13 " " 1 "" {INLPLOT "6(-%'CURVESG6%7S7$\"\"!$\"\"\"F(7$$\"3lFjF=g!>U#!#?$ \"3&3`Cg/bt+\"!#<7$$\"3r]u\"\\\\$>HXF.$\"3%[k-J?/R,\"F17$$\"39,vrc'e!* *oF.$\"3VM0L%>V9-\"F17$$\"3q6c9BGj%G*F.$\"31\"\\x o)e;\"!#>$\"3%*3gQ\"FD$\"3lO-GdctW5F17$$\"3W%Q;, fKRh\"FD$\"3(*eIps_w_5F17$$\"3bvd0`:l\\=FD$\"3*pNXGQ'Hh5F17$$\"3CQ:HiX h%3#FD$\"3,HVTyI FD$\"3TC9:cG]26F17$$\"3q=/B0jO^KFD$\"3P61)oRLv6\"F17$$\"3i*)*Qt*4&>Y$F D$\"3?q,8Oq#p7\"F17$$\"3nyNWklN7PFD$\"3q;YW(Gs%Q6F17$$\"3hxM`DH[CRFD$ \"3[_%)Qf<\"F17$$\"36//PXYHHYFD$\"3z8IFYQr%=\"F17$$\"3F&pe9Q)[d[FD$ \"35xTWe;L(>\"F17$$\"3'G#>(fg%e&4&FD$\"31XcC3)R5@\"F17$$\"3_!GscEJUJ&F D$\"371p&)=U:C7F17$$\"3$R%oHE72]bFD$\"3g')oQL,\"*Q7F17$$\"31ZBxjE/&z&F D$\"3LZqsKX'\\D\"F17$$\"3].DB?5H3gFD$\"3:=F)G5&fp7F17$$\"3Y-\"Rt30'QiF D$\"3&=V%>?<9'G\"F17$$\"3'3!z!\\0UlZ'FD$\"3)3z1!p:7/8F17$$\"3m+CmdpJ4n FD$\"3IBEkM[mA8F17$$\"3$)y)R^kQX$pFD$\"3%yKxTq!fT8F17$$\"34%4P)H*3Y=(F D$\"3TW2\"fN`QO\"F17$$\"3:sF'H?3$4uFD$\"3))f;+QD4&Q\"F17$$\"38mR7QEA\\ wFD$\"3a)*R#><)>49F17$$\"30+jnu2imyFD$\"3#*Rc9ftWK9F17$$\"3t\"o+.O%H/ \")FD$\"3b9[^_!f&f9F17$$\"3!edP?oEzK)FD$\"3<:HU:w&o[\"F17$$\"3!\\;x\\* ooh&)FD$\"36sWV@AX<:F17$$\"3%oX-1BI-z)FD$\"3f`A;7ci\\:F17$$\"3\"3;3F#R \\H!*FD$\"3a^BAq)zfe\"F17$$\"3e-.P&HL*f#*FD$\"3K1lC2f(Qi\"F17$$\"3,/M, P3f&\\*FD$\"3:&)Hg9.!fm\"F17$$\"3Bu2B:qHH(*FD$\"3$3X#RiuA6!=5Ty,>5!#=$\"3]nz#ot(*H\"=F17$$\"3 $3(*pR(>.T5Fby$\"3-i`(>AE$o=F17$$\"3i\\N9JO]k5Fby$\"3Y'p!o!obG$>F17$$ \"3UOY(>+pp3\"Fby$\"3^g!p9B#f+?F17$$\"3$******46666\"Fby$\"3RjX)*)fx13 #F1-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%&STYLEG6#%&POINTG-F$6%7UF'7$F ,$\"3jIX-Y]N25F1F27$F8$\"3wN0L%>V9-\"F17$F=$\"38YvT$\\E#H5F17$FB$\"31C _PNe#)=FFlF0\"F17$FR$\"3xEM'HQ'Hh5 F17$FW$\"37kY')y\\YYLv6\"F17$F_p$\"3 yYHgur#p7\"F17$Fdp$\"3d#R\\\\fs%Q6F17$Fip$\"3gc,'3M#f[6F17$F^q$\"3V1x0 GDyg6F17$Fcq$\"3'[>Id'R(><\"F17$Fhq$\"3o=')zkxr%=\"F17$F]r$\"3'=m\"F17$Fbr$\"34M5MG=067F17$Fgr$\"31mB8BR]D\"F17$Ffs$\"3#4')HM!*z'p7F17$F[t$\"3&>@r[2viG\"F17$F`t $\"36JRHW9L/8F17$Fet$\"3/Zra`x)HK\"F17$Fjt$\"3%H5![%Gv?M\"F17$F_u$\"3N #)o&\\.0YO\"F17$Fdu$\"3(*y)>$3r>'Q\"F17$Fiu$\"3#Q9@B&*[3T\"F17$F^v$\"3 .)fU,L0[V\"F17$Fcv$\"3@rb`P\\,j9F17$Fhv$\"3`Wvj\\ox\"\\\"F17$F]w$\"3HB tCFD_C:F17$Fbw$\"3G^:$G$ykf:F17$Fgw$\"37bM6T*R.g\"F17$F\\x$\"3kJ+7?&yS k\"F17$Fax$\"3)y'RG9$3Wp\"F17$Ffx$\"3AK1jma;^F17$F[z$\"3)>i^/[#4x?F17$$ \"3-$4flJOd2\"Fby$\"3O\\@Tj4=M@F17$F`z$\"3[N?#o#*fm>#F17$$\"3<=t[c+/*4 \"Fby$\"3hRlW*4#fqAF17$Fez$\"3Oc!pc#f]_BF1-Fjz6&F\\[lF(F][lF(F`[l-F$6% 7XF'7$F,$\"3jGX-Y]N25F17$F3$\"3iWE5.U!R,\"F17$F8$\"35N0L%>V9-\"F17$F=$ \"3oXvT$\\E#H5F17$FB$\"3tA_PNeOe'R(><\"F17$Fhq$\"3q]L8lxr%=\"F17$F]r$\"3T>$oR_Qt>\"F17$Fb r$\"33XG.J=067F17$Fgr$\"3.TK')HR]D\"F17$Ffs$\"3.'GwI+!op7F17$F[t$\"3%QGsQIviG\"F17$F`t$\"3Pq&*Gq> L/8F17$Fet$\"3B\\0W5*))HK\"F17$Fjt$\"3dx=16x2U8F17$F_u$\"3$eYr3T\"F17$F^v$\"3ixG\"fE[[V\"F1 7$Fcv$\"3G\"*)34p+JY\"F17$Fhv$\"3ibQ6%HR>\\\"F17$F]w$\"3BU%)p\"F17$Ffx$\"32AksLXyeF17$Ffy$\"3`cA=\"f4^-#F17$$\"3Agnb-yw_5Fby$\"3HDZ^2kz(4#F17$F[z$ \"3oBNw@Yz'=#F17$F_dl$\"3OaIUaj![H#F17$F`z$\"3^7$QG*>.QCF17$$\"3Ix4BHX +$4\"Fby$\"3i8n*\\5rq`#F17$Fgdl$\"3`-eC.*=&eEF17$$\"30fOu$ev]5\"Fby$\" 3B/0F<'>2\"GF17$Fez$\"3xMaU'zKe+$F1-Fjz6&F\\[lF(F(F][lF`[l-%&TITLEG6#% gnTaylor~series~(red=10~terms,~green=20~terms,~blue=100~terms)G-%+AXES LABELSG6$%\"zG%\"yG-%%VIEWG6$;F($\"+66666!#5;F($\"\"&F(" 2 574 574 574 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}}{SECT 1 {PARA 5 "" 0 "co mputing a basis" {TEXT -1 43 "Computing a basis of asymptotics expansi ons" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "To compute asymptotic expan sions at the singularity " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 54 ", we find it convenient to center the equation around " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 39 ". We thus make the change of variable \+ " }{XPPEDIT 18 0 "u=1-9*z" "/%\"uG,&\"\"\"\"\"\"*&\"\"*F&%\"zGF&!\"\" " }{TEXT -1 8 ", i.e., " }{XPPEDIT 18 0 "f(u)=j((1-u)/9)" "/-%\"fG6#% \"uG-%\"jG6#*&,&\"\"\"\"\"\"F&!\"\"F-\"\"*F." }{TEXT -1 6 " (and " } {XPPEDIT 18 0 "z=(1-u)/9" "/%\"zG*&,&\"\"\"\"\"\"%\"uG!\"\"F'\"\"*F)" }{TEXT -1 139 "). To perform this rational change of variable, we app eal to the closure of the class of holonomic functions under algebraic substitution:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "d_eq:=co llect(subs(\{j=f,z=u\},algebraicsubs(double_inv_borel(power_j(3),j,z), j-(1-z)/9,j(z))),diff,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%d _eqG,(**%\"uG\"\"\",&F'F(\"\")F(F(,&F'F(!\"\"F(F(-%%diffG6$-F.6$-%\"fG 6#F'F'F'F(F,*&,(F*F(F'!#9*$F'\"\"#!\"$F(F0F(F(*&,&F9F(F'F(F(F2F(F," }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Let us compute the indicial polyn omial at the singularity " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 12 " (now in 0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "indicial_p oly(\{d_eq\},f,u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$%$etaG\"\"#" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "Hence, we expect one regular sol ution at 0, and one singular with logarithmic behaviour." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "We therefore expect an analytic solution and a logarithmic solution. This is confirmed by the following forma l expansion:" }}}{EXCHG }{EXCHG }{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Order:=3: dsolve(d_eq,f(u),series);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"uG,&*&%$_C1G\"\"\"++F'F+\"\"!#F+\"\"%\" \"\"#\"\"&\"#K\"\"#-%\"OG6#F+\"\"$F+F+*&%$_C2GF+,&*&-%#lnGF&F+F,F+F++) F'#\"\"$\"\")\"\"\"#\"#L\"$G\"\"\"#F5\"\"$F+F+F+" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 129 "We compute but do not display the series for a la rger order, so as to increase the precision in the numerical calculati ons below." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Order:=15: ds ol_ser_f:=dsolve(d_eq,f(u),series):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Note that there exist algorithms to perform the previous calcul ation much more efficiently." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "T he following plot displays the asymptotic behaviours at " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 18 " of the solutions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 364 "plots[display](\n plot(subs(u=1-9*z,c onvert(subs(\{_C1=1,_C2=0\},op(2,dsol_ser_f)),polynom)),\n z=0. .1/5,y=0..5,color=blue),\n plot(subs(u=1-9*z,convert(subs(\{_C1=0,_ C2=-1\},op(2,dsol_ser_f)),polynom)),\n z=0..1/9,y=-2..5,color=g reen),\n plot([[1/9,-2],[1/9,5]],color=black),\ntitle=`Asymptotic b ehaviours at 1/9 (blue=analytic, green=logarithmic)`);" }}{PARA 13 "" 1 "" {INLPLOT "6(-%'CURVESG6$7V7$\"\"!$\"3_0,6z!H6,#!#<7$$\"3\\mmm;arz @!#?$\"3:z[!='QcB>F+7$$\"3(HLLL$3VfVF/$\"3(RXPXw!fY=F+7$$\"3q****\\i&* )fD'F/$\"3`Ejm\"oJqy\"F+7$$\"3Hnmm\"H[D:)F/$\"3Jf'[(>dXL$\"3j]`Jh\">%z;F+7$$\"3SLLLe0$=C\"FD$\"3w@'*p,-3J;F+7$$\"3ILLL 3RBr;FD$\"3J$QKv/&)za\"F+7$$\"3hmm;zjf)4#FD$\"3m0s=P#f,[\"F+7$$\"3LLL$ e4;[\\#FD$\"3MQ%QW]dtU\"F+7$$\"3$)****\\i'y]!HFD$\"3!G1(4&\\$e!Q\"F+7$ $\"3:LL$ezs$HLFD$\"3)[7$Q!HN)Q8F+7$$\"3$*****\\7iI_PFD$\"3$*y8(*y7_-8F +7$$\"3?nmm;_M(=%FD$\"3QQEN`\"z&p7F+7$$\"3_LLL3y_qXFD$\"3#[s3d<)eV7F+7 $$\"3X+++]1!>+&FD$\"3#QTc*)pJr@\"F+7$$\"3f******\\Z/NaFD$\"3N!oRW9sI> \"F+7$$\"3&*******\\$fC&eFD$\"3SF'p2R$*=<\"F+7$$\"3ELL$ez6:B'FD$\"3\"4 ]>+pGT:\"F+7$$\"36mmm;=C#o'FD$\"3[,l?r6eM6F+7$$\"37nmmm#pS1(FD$\"3bYWm 4\")>>6F+7$$\"3y****\\i`A3vFD$\"3\"=BgL!o\\-6F+7$$\"3Wlmmm(y8!zFD$\"3X y7FCgm)3\"F+7$$\"3:++]i.tK$)FD$\"3'[g=>14W2\"F+7$$\"39++](3zMu)FD$\"3( 3nfNmN;1\"F+7$$\"3>nmm\"H_?<*FD$\"3q0=&zBg!\\5F+7$$\"3dmm;zihl&*FD$\"3 -k;\"=mE\"Q5F+7$$\"3VLLL3#G,***FD$\"3U\\c0cV$p-\"F+7$$\"3GLLezw5V5!#=$ \"3!)=kZyB\"f,\"F+7$$\"3)****\\PQ#\\\"3\"Fbt$\"3vi.Setx15F+7$$\"3MLL$e \"*[H7\"Fbt$\"3P>b3%)4at**Fbt7$$\"3!*******pvxl6Fbt$\"3.$o**))=d1))*Fb t7$$\"3'*****\\_qn27Fbt$\"3*fp#z*ebQz*Fbt7$$\"3$****\\i&p@[7Fbt$\"3)y; Nh5EMr*Fbt7$$\"3$*****\\2'HKH\"Fbt$\"3>BfgOD$zi*Fbt7$$\"3UmmmwanL8Fbt$ \"3m8C+'\\-Vb*Fbt7$$\"38+++v+'oP\"Fbt$\"3f#fiUD)zy%*Fbt7$$\"3CLLeR<*fT \"Fbt$\"3NF%)ypX)HT*Fbt7$$\"3-+++&)Hxe9Fbt$\"3U\\m8uOpV$*Fbt7$$\"3Ymm \"H!o-*\\\"Fbt$\"3#\\DP6nI$))Fbt7$$\"3? 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ABELSG6$%\"zG%\"yG-%%VIEWG6$;F($\"+++++?!#5;F`_mFc_m" 2 574 574 574 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "P[3](z)" "-&%\"PG6#\"\"$6#%\"zG " }{TEXT -1 44 " is a linear combination of both behaviours." }}}} {SECT 1 {PARA 5 "" 0 "the connection" {TEXT -1 14 "The connection" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "To perform the connection, we dete rmine values for " }{XPPEDIT 18 0 "_C1" "I$_C1G6\"" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "_C2" "I$_C2G6\"" }{TEXT -1 15 " above so that " } {XPPEDIT 18 0 "dsol_ser_f" "I+dsol_ser_fG6\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "ser_j[precise]" "&%&ser_jG6#%(preciseG" }{TEXT -1 12 " \+ agree when " }{XPPEDIT 18 0 "z" "I\"zG6\"" }{TEXT -1 21 " takes values around " }{XPPEDIT 18 0 "1/18" "*&\"\"\"\"\"\"\"#=!\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=25:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "We graphically determine " } {XPPEDIT 18 0 "_C1" "I$_C1G6\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 153 "plot(subs(_C1=x,map(solve,\{seq(series_j(50,( 1-(.5+.01*i))/9)=eval(subs(u=.5+.01*i,convert(op(2,dsol_ser_f),polynom ))),i=1..9)\},_C2)),x=0.85984.. .85986);" }}{PARA 13 "" 1 "" {INLPLOT "6--%'CURVESG6$7S7$$\"&%)f)!\"&$!96)G^lr@@E[]8%!#C7$$\":LLLLL$3VfVS)f) !#D$!9w\"\\))p(fL%QO]8%F-7$$\":nmmm;H[D:3%)f)F1$!9-O,<:\"Q&\\g-NTF-7$$ \":LLLLLe0$=CT)f)F1$!9*o1*yOq(pU9]8%F-7$$\":LLLLL3RBr;%)f)F1$!9d415zSP FF+NTF-7$$\":nmmm;zjf)4U)f)F1$!9e3#3?HxL3\"*\\8%F-7$$\":LLLL$e4;[\\U)f )F1$!9ceS.N+X)f)F1$!9T)**fD28%z>\"\\8%F- 7$$\":+++++]Z/Na%)f)F1$!9oe?p\\*))y<+\\8%F-7$$\":+++++]$fC&e%)f)F1$!9E Fg1)oL\\!)))[8%F-7$$\":LLLL$ez6:BY)f)F1$!9H]R#>w+sZy[8%F-7$$\":nmmmm;= C#oY)f)F1$!9R6e-$RFl>m[8%F-7$$\":nmmmmm#pS1Z)f)F1$!9P58*4>'=$zb[8%F-7$ $\":++++]i`A3v%)f)F1$!9pW0V'>F;pV[8%F-7$$\":nmmmmm(y8!z%)f)F1$!9O3])f& 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"fsol:=\{_C1=.85985,_C2=convert(\",`+`)/nops(\")\};" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%fsolG<$/%$_C1G$\"&&)f)!\"&/%$_C2G$ !:X&3Y.>c--@xMT!#D" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Plugging th e previous values into " }{XPPEDIT 18 0 "dsol_ser_f" "I+dsol_ser_fG6\" " }{TEXT -1 56 " yields the following truncated asymptotic expansion o f " }{XPPEDIT 18 0 "P[3](z)" "-&%\"PG6#\"\"$6#%\"zG" }{TEXT -1 1 ":" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "subs(fsol,collect(subs(u= 1-9*z,convert(series(op(2,dsol_ser_f),u,3),polynom)),[ln(1-9*z),z],nor mal));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,**&,(*$%\"zG\"\"#$!:Skn!4uh iT42L_!#CF'$\":w0FO'p/lw$GK4#F+$!:m#=uwENZoK_9e!#D\"\"\"F1-%#lnG6#,&F1 F1F'!\"*F1F1F&$\":uQQ]F\"o\")3&4zC#F+F'$!:)erDu*3q>O7(Q5F+$\":!yhY&H\" p=a^5v%*F0F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Once again, we co mpute a more precise expansion, but do not display it." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "j_num:=subs(fsol,collect(subs(u=1-9 *z,convert(op(2,dsol_ser_f),polynom)),[ln(1-9*z),z],normal)):" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Plotting " }{XPPEDIT 18 0 "P[3](z) " "-&%\"PG6#\"\"$6#%\"zG" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "We hav e three means to evaluate numerical values of " }{XPPEDIT 18 0 "P[3](z )" "-&%\"PG6#\"\"$6#%\"zG" }{TEXT -1 94 ": its Taylor series at 0, the differential equation it satisfies, the asymptotic expansion at " } {XPPEDIT 18 0 "1/9" "*&\"\"\"\"\"\"\"\"*!\"\"" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "The following plot is the numerica lly unstable result of following the graph of " }{XPPEDIT 18 0 "P[3]" "&%\"PG6#\"\"$" }{TEXT -1 103 " by using the differential equation. T he method used is the fourth-order classical Runge-Kutta method." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 221 "pl[RungeKutta]:=subs(\{LINE =POINT,COLOR(RGB,.9,.9,.2)=COLOR(RGB,0,0,0)\},DEtools[DEplot](remove(t ype,double_inv_borel(power_j(3),j,z),equation),j(z),z=0.0001.. 1/9,[[j (.0000001)=1,D(j)(.0000001)=3]],j=0..5,stepsize=.0005)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "The following plot is the result of the c onnection." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "pl[connection ]:=plot(j_num,z=0..1/9,y=0..5,style=point,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "plots[display](eval(subs(COLOUR=pr oc() COLOUR(RGB,1.,0,0) end,pl[Taylor])),pl[RungeKutta],pl[connection] ,title=`black=Runge-Kutta, red=Taylor, green=connection`);" }}{PARA 13 "" 1 "" {INLPLOT "6*-%'CURVESG6%7S7$\"\"!$\"\"\"F(7$$\"3lFjF=g!>U#! #?$\"3&3`Cg/bt+\"!#<7$$\"3r]u\"\\\\$>HXF.$\"3%[k-J?/R,\"F17$$\"39,vrc' 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More specifically, the best approximation by the Tayl or series is obtained for a truncation of order 100, whereas the expan sion obtained by connection is of order 15 only." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 53 "We are now able to give an equivalent for the numb er " }{XPPEDIT 18 0 "p[3,n]" "&%\"pG6$\"\"$%\"nG" }{TEXT -1 84 " of pa irs of paths with same origin and end. Retaining only the singular pa rt when " }{XPPEDIT 18 0 "u" "I\"uG6\"" }{TEXT -1 14 " tends to 0 of" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Order:=3: dsolve(d_eq,f(u ),series);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"uG,&*&%$_C1G \"\"\"++F'F+\"\"!#F+\"\"%\"\"\"#\"\"&\"#K\"\"#-%\"OG6#F+\"\"$F+F+*&%$_ C2GF+,&*&-%#lnGF&F+F,F+F++)F'#\"\"$\"\")\"\"\"#\"#L\"$G\"\"\"#F5\"\"$F +F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "we get the equivalent " }{XPPEDIT 18 0 "_C2*ln(1-9*z)" "*&%$_C2G\"\"\"-%#lnG6#,&\"\"\"F$*&\"\" *F$%\"zGF$!\"\"F$" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "P[3](z)" "-&%\"P G6#\"\"$6#%\"zG" }{TEXT -1 22 ". An asymptotics for " }{XPPEDIT 18 0 "p[3,n]" "&%\"pG6$\"\"$%\"nG" }{TEXT -1 12 " follows by " }{TEXT 284 23 "transfer of singularity" }{TEXT -1 24 ": an asymptotic for the " } {XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 63 "-th coefficient a series is computed using the following table:" }}}{EXCHG {PARA 0 "> " 0 "transf er of singularity" {MPLTEXT 1 0 308 "matrix([[Function,`Asymptotic coe fficient`,` `],[(1-z/rho)^(-alpha)*ln(1-z/rho)^beta,rho^n*n^(alpha-1)* ln(n)^beta/GAMMA(alpha),``(beta>=0 and alpha*` non integer`)],[(1-z/rh o)^(-p)*ln(1-z/rho)^beta,rho^n*n^(p-1)*ln(n)^(beta-1)/GAMMA(alpha),``( beta>=0 and p*` integer`)],[1/ln(1-z/rho),rho^n/n/ln(n)^2,` `]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7%%)FunctionG%7Asymptoti c~coefficientG%\"~G7%*&),&\"\"\"F/*&%\"zGF/%$rhoG!\"\"F3,$%&alphaGF3F/ )-%#lnG6#F.%%betaGF/**)F2%\"nGF/)F=,&F5F/F3F/F/)-F86#F=F:F/-%&GAMMAG6# F5F3-%!G6#31,$F:F3\"\"!*&F5F/%-~non~integerGF/7%*&)F.,$%\"pGF3F/F6F/** F " 0 "" {MPLTEXT 1 0 35 "asympt_p[3 ]:=-subs(fsol,_C2)*9^n/n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)asymp t_pG6#\"\"$,$*&)\"\"*%\"nG\"\"\"F,!\"\"$\":X&3Y.>c--@xMT!#D" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 16 "Asymptotics for " }{XPPEDIT 18 0 "Q[3](z)" "-&%\"QG6#\"\"$6#%\"zG" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "We now turn to our original goal: evaluating the coefficients of" }}}{EXCHG {PARA 274 "" 0 "" {XPPEDIT 18 0 "Q[3](z)=(1+3*z-1/P[3](z))/2 " "/-&%\"QG6#\"\"$6#%\"zG*&,(\"\"\"\"\"\"*&\"\"$F-F)F-F-*&\"\"\"F--&% \"PG6#\"\"$6#F)!\"\"F8F-\"\"#F8" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "It follows from the value of " }{XPPEDIT 18 0 "P[3]( z)" "-&%\"PG6#\"\"$6#%\"zG" }{TEXT -1 39 " that the ordinary generatin g function " }{XPPEDIT 18 0 "Q[3](z)" "-&%\"QG6#\"\"$6#%\"zG" }{TEXT -1 14 " behaves like " }{XPPEDIT 18 0 "2/3-1/(2*alpha*ln(1-9*z))" ",&* &\"\"#\"\"\"\"\"$!\"\"F%*&\"\"\"F%*(\"\"#F%%&alphaGF%-%#lnG6#,&\"\"\"F %*&\"\"*F%%\"zGF%F'F%F'F'" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "z" "I \"zG6\"" }{TEXT -1 13 " is close to " }{XPPEDIT 18 0 "1/9" "*&\"\"\"\" \"\"\"\"*!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "alpha=.413477" "/% &alphaG$\"'xMT!\"'" }{TEXT -1 52 ". Because of the positivity of the \+ coefficients of " }{XPPEDIT 18 0 "P[3](z)" "-&%\"PG6#\"\"$6#%\"zG" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "P[3](z)" "-&%\"PG6#\"\"$6#%\"zG" } {TEXT -1 39 " never vanishes before its singularity " }{XPPEDIT 18 0 " s=1/9" "/%\"sG*&\"\"\"\"\"\"\"\"*!\"\"" }{TEXT -1 10 ", so that " } {XPPEDIT 18 0 "Q[3](z)" "-&%\"QG6#\"\"$6#%\"zG" }{TEXT -1 32 " also ha s no singularity before " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 33 " , as visualized on the next plot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plot(2/3-1/2/j_num,z=0..1/9,y=-.1..1,style=point,colo r=green);" }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6#7gn7$\"\"!$\":_y $zIMdR*Q-_h\"!#D7$$\":>&=&=&=&=&=g!>U#!#F$\":(znU?Ip,)=%[m;F+7$$\":r.P q.Pq`\\$>HXF/$\":J>[O=F+7$$\":I'H'H'H'H@%*3gQ\"FD$\":w(ep9*>ynZ-T (=F+7$$\":yxxxxxx-fKRh\"FD$\":HZnI\")GsAFYE\">F+7$$\":T2uS2uSKb^'\\=FD $\":B&3sUmz,xVJ_>F+7$$\":+++++++Dc9Y3#FD$\":EGq!z^Q,NF&=*>F+7$$\":[\"[ \"[\"[\"[\")*GIEBFD$\":]6/6lD7k_xE.#F+7$$\":T2uS2uSd6#=RDFD$\":Ty1&)RZ 2IAl)o?F+7$$\":XWWWWWWWpL)yFFD$\":(G@^d1X+q!\\*4@F+7$$\":cbbbbbbbIp%>I FD$\":XKd]G,2.+X;:#F+7$$\":cbbbbbbbIm8D$FD$\":v4V/4*))G(e3B>#F+7$$\":& =&=&=&=&o(*4&>Y$FD$\":4&Gm%4Q)[d[FD$\":n' p*p*R2Zx(R2\\#F+7$$\":#[\"[\"[\"[\"[1Ye&4&FD$\":5q#z>T\">KzC!QDF+7$$\" :/Pq.Pq.iEJUJ&FD$\":/f,#=e(*)z=yAe#F+7$$\":_=&=&=&=&oAr+b&FD$\":f)GY1Q f/!>k4j#F+7$$\":_=&=&=&=NkE/&z&FD$\":EovY)exM:rl#o#F+7$$\":MLLLLLL3-\" H3gFD$\":t+quW\"41.`lGFF+7$$\":jH'H'H'H'z30'QiFD$\":h]g\"zz]Df#f%zFF+7 $$\":cbbbbbbb0UlZ'FD$\":*=\"H>'=r*3coK$GF+7$$\":MLLLLLL$epJ4nFD$\":GS! =iolw3SK()GF+7$$\":MLLLLLLekQX$pFD$\":%*Qj8dhc(4a2THF+7$$\":cbbbbbb0$* 3Y=(FD$\":)3rVA2w#fT*e-IF+7$$\":/Pq.Pq.P?3$4uFD$\":#**=C\")HQzpNnfIF+7 $$\":!*)))))))))))))QEA\\wFD$\":J?aqDXh**4CF7$F+7$$\":kH'H'H'HYv2imyFD $\":IC#\\PM#H<^T>=$F+7$$\":76666666O%H/\")FD$\":R]+i=;]<6G#\\KF+7$$\": r.Pq.PqGoEzK)FD$\":d6Gk^*>HeFG:LF+7$$\":MLLLLLLe*ooh&)FD$\":\">O$eqa,R z\"e(Q$F+7$$\":#[\"[\"[\"[\"[J-B!z)FD$\":]R+-@TR&Q@)>Y$F+7$$\":7666666 O#R\\H!*FD$\":.Sj5YknBk,Xa$F+7$$\":(H'H'H'H'H'HL*f#*FD$\":QotU:;$)*oCL HOF+7$$\":kH'H'H'H'z$3f&\\*FD$\":&)\\p2j%*\\g;TFs$F+7$$\":/Pq.Pq.i,(HH (*FD$\":Vwp6(G>Ar7uBQF+7$$\":BAAAAAAAs[S%**FD$\":2NP6$Q&eGjog#RF+7$$\" :/Pq.Pq.7%y,>5F+$\":#e\">z;xgF88(eSF+7$$\":uS2uS2uS(>.T5F+$\":*4vZWj)f s30x>%F+7$$\":++++++]7j.X1\"F+$\":agU_-_.Pz%G\"Q%F+7$$\":nmmmmmmmJOd2 \"F+$\":I[*od+soA8\\\"\\%F+7$$\":LLLLLL$3-!pp3\"F+$\":z,Rl]CfPvt%GYF+7 $$\":yxxxxFS$HX+$4\"F+$\":%yout;!)>#['=@ZF+7$$\":AAAAAA(fc+/*4\"F+$\": UiF-vvP2+R)Q[F+7$$\":XWWWWpD-#y0-6F+$\":x\"p#pg<#po!oU\"\\F+7$$\":nmmm m;aQev]5\"F+$\":A3GpRo,lC53,&F+7$$\":yxxxFSocY%e16F+$\":*GqBFW/8#>8K2& F+7$$\":*))))))))QE[ZL436F+$\":&p/LD=@L[5p`^F+7$$\":XWWWpv*Q)yZ)36F+$ \":#o- " 0 "" {MPLTEXT 1 0 47 "asympt_q[3]:=-subs(fsol,1/_C2)*9^n/n/ln(n)^2/2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)asympt_qG6#\"\"$,$*()\"\"*%\"nG\" \"\"F,!\"\"-%#lnG6#F,!\"#$\":6\\7Pf'3B!\\c#47!#C" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 38 "Numerical check of the asymptotics of " } {XPPEDIT 18 0 "p[3,n]" "&%\"pG6$\"\"$%\"nG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "q[3,n]" "&%\"qG6$\"\"$%\"nG" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "To get satisfactory results, we have to work for large " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "NMax:=250;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% %NMaxG\"$]#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "We create another \+ procedure to compute the coefficients of the series " }{XPPEDIT 18 0 " P[3](z)" "-&%\"PG6#\"\"$6#%\"zG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 89 "proc_p:=rectoproc(diffeqtorec(double_inv_borel (power_j(3),j,z),j(z),u(n)),u(n),remember);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'proc_pG:6#%\"nG6\"6#%)rememberGE\\s%\"\"\"\"\"$\"\"! F,\"\"#\"#:F-\"#$**&,(-9!6#,&9$F,!\"#F,!\"*-F56#,&F8F,!\"\"F,F-*&,(F4 \"#=F;!#5*&,&F4F:F;\"#5F,F8F,F,F,F8F,F,F,F8F9F(F(" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 31 "We get a precise expansion for " }{XPPEDIT 18 0 "P [3](z)" "-&%\"PG6#\"\"$6#%\"zG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 54 "series_p:=convert([seq(proc_p(i)*z^i,i=0..NMax )],`+`):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "from which follows a \+ precise expansion for " }{XPPEDIT 18 0 "Q[3](z)" "-&%\"QG6#\"\"$6#%\"z G" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "series _q:=series(1/2+3*z/2-1/series_p/2,z,NMax+1):" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 28 "The asymptotic behaviour of " }{XPPEDIT 18 0 "p[3,n]" " &%\"pG6$\"\"$%\"nG" }{TEXT -1 20 " is easily attained." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for i from 50 to NMax by 50 do i=ev alf(subs(n=i,asympt_p[3])/coeff(series_p,z,i)) od;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/\"#]$\":PfXLK6*f;B&\\+\"!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+\"$\":QfG`F$>kpFX-5!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$]\"$\":>ijOk%p*)3&>;+\"!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+#$\":P\"H\"Hz,M*pG?,5!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$]#$\":%>@tzYqd%)G&4+\"!#C" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 17 "The behaviour of " }{XPPEDIT 18 0 "q[3,n]" "&%\"qG6$\" \"$%\"nG" }{TEXT -1 79 ", however, is not observed so obviously, due t o a slow logarithmic convergence." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for i from 50 to NMax by 50 do i=evalf(subs(n=i,asymp t_q[3])/coeff(series_q,z,i)) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ \"#]$\":hwdc8R*[g_DOK!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+\"$\" :,eUX!QY%R%=U&y#!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$]\"$\":3w8V T.5&oYW#f#!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+#$\":!)ovJkK*=]T dxC!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$]#$\"::9SU_=63xL')R#!#C " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Enlarging " }{XPPEDIT 18 0 "n " "I\"nG6\"" }{TEXT -1 49 ", of course, would lead to a limiting value of 1." }}}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Conclusion" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "To conclude, we compute the table indicating the indicial polynomial at the dominant singularity and th e possible asymptotic behaviours of " }{XPPEDIT 18 0 "P[d](z)" "-&%\"P G6#%\"dG6#%\"zG" }{TEXT -1 7 " there." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "behaviours:=proc(f,x,u) local L,i;\n L:=sort([sol ve(f,x)]);\n convert([seq(u^L[i]*ln(u)^nops(select(type,map(`+`,L[1 ..i-1],-L[i]),integer)),i=1..nops(L))],`+`)\nend:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 71 "M[1]:=d,s,`indicial polynomial at s`,`possib le behaviours for P[d](z)`:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 189 "for i from 2 to 7 do\n M[i]:=i,1/i^2,\n indicial_poly( \{algebraicsubs(double_inv_borel(power_j(i),j,z),j=(1-z)/i^2,j(z))\},j ,z);\n B:=behaviours(M[i][3],eta,u);\n M[i]:=M[i],B\nod:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "matrix([seq([M[i]],i=1..7)]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7)7&%\"dG%\"sG%9indic ial~polynomial~at~sG%@possible~behaviours~for~P[d](z)G7&\"\"##\"\"\"\" \"%,&F/F/%$etaGF-*$%\"uG#!\"\"F-7&\"\"$#F/\"\"**$F2F-,&F/F/-%#lnG6#F4F /7&F0#F/\"#;,$*(F2F/,&F2F-F6F/F/,&F2F/F6F/F/F6,(F/F/*$F4#F/F-F/*&F4F/F =F/F/7&\"\"&#F/\"#D,$*(F2F/,&F2F/!\"#F/F/FFF-F6,*F/F/FJF/*&F4F/F=F-F/* &F4F-F=F8F/7&\"\"'#F/\"#O*,F2F/FFF/FQF/,&F2F/!\"$F/F/,&F2F-FfnF/F/,,F/ F/FJF/*$F4#F8F-F/*&F4F-F=F-F/*&F4F8F=F8F/7&\"\"(#F/\"#\\*,F2F/FFF/FenF /,&F2F/!\"%F/F/FQF-,.F/F/FJF/F[oF/FUF/*&F4F8F=F0F/*&F4F0F=FLF/" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "P[d ](z)" "-&%\"PG6#%\"dG6#%\"zG" }{TEXT -1 146 " is a linear combination \+ of the possible behaviours. Solving the connection would indicate whi ch of them appear (with a non-zero coefficient) in " }{XPPEDIT 18 0 "P [d](z)" "-&%\"PG6#%\"dG6#%\"zG" }{TEXT -1 9 ". Next, " }{HYPERLNK 17 "transfer of singularity" 1 "" "transfer of singularity" }{TEXT -1 62 " on the dominant behaviour would yield an asymptotic form for " } {XPPEDIT 18 0 "p[d,n]" "&%\"pG6$%\"dG%\"nG" }{TEXT -1 1 "." }}}}} {MARK "3" 0 }{VIEWOPTS 1 1 0 1 1 1803 } &F/F/%$etaGF-*$%\"uG#!\"\"F-7&\"\"$#F/\"\"**$F2F-,&F/F/-%#lnG6#F4F /7&F0#F/\"#;,$*(F2F/,&F2F-F6F/F/,&F2F/F6F/F/F6,(F/F/*$F4#F/F-F/*&F4F/F =F/F/7&\"\"&#F/\"#D,$*(F2F/,&F2F/!\"#F/F/FFF-F6,*F/F/FJF/*&F4F/F=F-F/* &F4F-F=F8F/7&\"\"'#F/\"#O*,F2F/FFF/FQF/,&F2F/!\"$F/F/,&F2F-FfnF/F/,,F/ F/FJF/*$F4#F8F-F/*&F4F-F=F-F/*&F4F8F=F8F/7&\"\"(#F/\"#\\*,F2F/FFF/FenF /,&F2F/!\"%F/F/FQF-,.F/F/FJF/F[oF/FUF/*&F4F8F=F0F/*&F4F0F=FLF/" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "P[d ](z)" "-&%\"PG6#%\"dG6#%\"zG" }{TEXT -1 146 " i<numgfunbounddiffeqtailpowerseriexpansfinitfunctcallsequenceqparameterlineardifferentialequatwithpolynomialcoefficientinitialvaluatoriginnamevariablstartindexdescriptcommandcomputsuminfinitanalyticgivensolutalongoutputformulainvolvsummatrangexperimentalexamplgfundiffalsoQqdocumentgeneratordeclaretextkeyworddeclarnewkeywordtextmadformatcallsequencparamnmadargoptparametersymbollisttypeprocedurintegdescriptdesignextendsyntaxconventnamestartnonnumbargumentspecifisuchcancheckactualcodedefinaddedtranslatortablexamplmylistmeantrepresobjectbooleanallowdisplabulletnowprovidimplementatlatextranslatetextproclocalopfalselsefistringbeginitemizseqitemendtestexportdocumdocumentclasamsbookusepackagamsmathamsopngraphicxspecialcommandnewcommandmaderrorflushleftfboxminipagtextwidthttmadierrorstaticheaddynamictruealsoweenconsistbecomremainpreviouorientatmarksignwaydenotordinargeneratfunctorderprecisesincanyviewequivalentcomputsingularitsmallexplicityieldclosformalreadusualhandusingvandermondconvolutreadimaplsumbinomialbecausdeterminchoicdirectthusgivencentralcoefficinormalconvertfactorialexpandnumberwellopinfinitanothcoefficientgottennewtonexpansasymptsimplifsubscospipolynomsymbolicvalufoundgeteasiintodeltakronecksymbolotherwisevaluatcorroboratseqcheckgfunfirstequatsatisfidiffevalsqrtrecursdiffeqtorecnextreturnprocedurefficicoeffrectoprocfinalourratiogoesdoevalfodlessspachighnolongsolutnaturaltoolattackbesseldefininstancconstantrespectalgebraicbelonglargclasholonomiccallsatisflinearrationalbenefitnumerouclosurpropertalgorithmhaveimplementunderborelinverstransformformalapplipowerenoughusrevisitformularecurrenceqrectodiffeqprocoptionremembpoltodiffeqsystemnowdoublinvsysinvboreldiffeqdsolvleadmethodanalysireadexponentialbehavioursectremovtypeperformcalculatsortexperimentalsuggestpatternspecificaldescribintroducparasiticalsoexpresstermmodifibothsolutionmycombinatabledevelopmsuccessiveplugunappbesselibesselkprovvanishatthosnicerepresentatnamealwayconverglowerboundhowevintegrandbehavlikeeithconjecturtheoremhomogeneouconnectanalyticneighbourhoudtaylorotherdominantclosestpriorismallestusealgorithmicalbasinumericalgenericpointcommonregularpolynomialexpectthereforlookundeterminexponindicialzeromultiplicitindicatpolylocalglobaletacollectfactorprimpartldegreremarknoteletlogarithmicsayrecentaroundstudtruncatactualneedseveralillustratslowconvergencbaseidentitalgebraicsubdividextractrecassociatparamincreasplstylcolorredbluefindconvenicentmakechangvariablappealsubstituthencconfirmprecisbelowdsolsermuchefficientdisplayblackagretakedigitgraphicalmapfsolnopsplugglnonceagainnumplottmeanunstablgraphfourthrungkuttarungekuttargbdetooldeplotstepsizcolourwhilrathcurvbestwhereagiveequivalretainparttendtransfthtablmatrixrhoalphabetagammaturnoriginalgoalpositivitneverbeforvisualcorrespondingsatisfactorworknmaxcreatattainobservobviousdueenlargcourslimitconclusconcludselectappearoremhomogeneouconnectanalyticneighbourhoudtaylorotherdominantclosestpriorismallestusealgorithmicalbasinumericalgenericpointcommonregularpolynomialexpectthereforlookundeterminexponindicialzeromultiplicitindicatpolylocalglobaletacollectfactorprimpartldegreremarknoteletlogarithmicsayrecentaroundstudtruncatactualneedseveralillustratslowconvergencbaseidentitalgebraicsubdividextractrecassociatparamincreasplstylcolorredbluefindconvenicentmakechangvariablappealsubstituthencconfirmprecisbelowdsolsermuchefficientdisplayblackagretakedigitgraphicalm{VERSION 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0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Dash \+ Item" 0 16 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 16 3 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 257 46 "A PROBLEM IN STATISTICAL CLASS IFICATION THEORY" }}{PARA 263 "" 0 "" {TEXT 283 0 "" }}{PARA 257 "" 0 "" {TEXT 276 17 "Philippe Flajolet" }{TEXT -1 1 " " }}{PARA 260 "" 0 " " {TEXT -1 29 "(Version of January 14, 1997)" }}{PARA 265 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "This problem discussed he re is at the origin of the whole " }{HYPERLNK 17 "Combstruct" 2 "comb struct" "" }{TEXT -1 108 " package. On October 8, 1992, Bernard Van Cu tsem, a statistician at the University of Grenoble wrote to us:\n" }} {PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 258 250 "In classification t heory, we make use of hierarchical classification trees. I would need \+ to generate at random such classification trees according to the unifo rm law. The elements to be classified may be taken as distinguished in tegers say from 1 to " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT 277 46 ". \+ Do you know of an algorithm for doing this?\n" }}{PARA 0 "" 0 "" {TEXT -1 174 "This led to a cooperation involving Paul Zimmermann, Ber nard Van Cutsem, and Philippe Flajolet, out of which the general theor y and the algorithms of Combstruct evolved, see " }{TEXT 259 28 "Theor etical Computer Science" }{TEXT -1 107 ", vol. 132, pp. 1-35. A first \+ implementation was designed by Paul Zimmermann in 1993, under the name Gaia (" }{TEXT 260 26 "Maple Technical Newsletter" }{TEXT -1 553 ", 1 994 (1), pp. 38-46).\n\nVan Cutsem's original question was motivated b y the following problem: Classification programmes in statistics build classification trees, usually proceeding by successive aggregations o f closest neighbours amongst existing classes. How can we measure the \+ way a classification carries useful information and not just \"random \+ noise\"? Certainly, \"good\" classification trees should exhibit chara cteristics that depart significantly from random ones. Hence the need \+ to simulate and analyse parameters of random classification trees." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 "S tatistics and classification theory" }}{SECT 1 {PARA 0 "" 0 "" {TEXT 256 13 "Specification" }}{PARA 0 "" 0 "" {TEXT -1 43 "We start by load ing the combstruct package." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "with(combstruct);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.%+allstruct sG%&countG%%drawG%)finishedG%'gfeqnsG%)gfseriesG%(gfsolveG%,iterstruct sG%+nextstructG%,prog_gfeqnsG%.prog_gfseriesG%-prog_gfsolveG" }}} {PARA 0 "" 0 "" {TEXT -1 140 " A classification is either: 1) an atom; 2) a set of classification trees of degree at least 2. Atoms are dist iguishable, hence we are in a " }{HYPERLNK 17 "labelled" 2 "combstruct [specification]" "" }{TEXT -1 25 " universe. Note that the " } {HYPERLNK 17 "Set" 2 "combstruct[specification]" "" }{TEXT -1 105 " co nstruction translates a pure graph-theoretic structure with no orderin g between descendents of a node." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "hier:=[H,\{H=Union(Z,Set(H,card>1))\},labelled]:" }}}{PARA 0 " " 0 "" {TEXT -1 68 "The original problem of Van Cutsem is solved by si ngle commands like" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "draw(h ier,size=10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SetG6%-F$6$-F$6$-F $6%&%\"ZG6#\"\"'&F-6#\"\"#-F$6$&F-6#\"\"%&F-6#\"\")&F-6#\"\"(&F-6#\"\" \"-F$6%&F-6#\"\"&&F-6#\"\"$&F-6#\"#5&F-6#\"\"*" }}}{PARA 0 "" 0 "" {TEXT -1 50 "We may adopt a more concise representation format:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "lreduce:=proc(e) eval(subs( \{Set=proc() \{args\} end, Sequence=proc() [args] end\},e)) end:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "lreduce(draw(hier,size=20)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$<$<$<$<$<$<$<$<$<%<$<$<$<$<$<$&% \"ZG6#\"\"*&F46#\"#8<$&F46#\"\"(&F46#\"#<&F46#\"#9<$&F46#\"\"#&F46#\"# ;&F46#\"#?&F46#\"\"'&F46#\"#:&F46#\"#>&F46#\"\"$&F46#\"#7&F46#\"#5&F46 #\"\"&&F46#\"\")&F46#\"#=&F46#\"#6&F46#\"\"\"&F46#\"\"%" }}}{PARA 0 " " 0 "" {TEXT -1 137 "Random generation takes only a few seconds while \+ counting tables (that serve to determine splitting probabilities) are \+ set up on the fly." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "for j \+ from 20 by 20 to 100 do j,lreduce(draw(hier,size=j)) od;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$\"#?<%<%&%\"ZG6#\"\"'&F'6#\"\"$&F'6#\"#8<$<$<$&F '6#\"\"(&F'6#\"#6&F'6#\"#;<$<$<$<$&F'6#\"\"#&F'6#\"#=&F'6#\"#<<$<$<$&F '6#\"\"%&F'6#F#&F'6#\"#5<%<$<%&F'6#\"\"&&F'6#\"#>&F'6#\"#7&F'6#\"\"*&F '6#\"\")&F'6#\"\"\"&F'6#\"#:&F'6#\"#9" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$\"#S<$<%&%\"ZG6#\"#O<%&F'6#\"#:&F'6#\"#;<$<$&F'6#\"#5&F'6#\"#I<$&F' 6#\"\"#<%&F'6#\"#?&F'6#\"#J<$&F'6#\"#6<$&F'6#\"#D<$<$&F'6#\"\"%&F'6#\" #7<%&F'6#\"\"&<$<$&F'6#\"#B&F'6#\"#L<$&F'6#\"\"\"&F'6#\"#M<$&F'6#\"#H< $&F'6#\"#<<$&F'6#\"#G<&&F'6#\"\"$&F'6#\"#F&F'6#\"#@<%&F'6#\"\")&F'6#\" \"*&F'6#\"#E<$&F'6#\"#C<$&F'6#\"#A<$&F'6#\"#N<$<$&F'6#F#&F'6#\"#P<$<$& F'6#\"#>&F'6#\"#K<$&F'6#\"\"'<%&F'6#\"\"(&F'6#\"#R&F'6#\"#Q<%&F'6#\"#9 &F'6#\"#8&F'6#\"#=" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$\"#g<%<$&%\"ZG6# \"\"*<$<$&F'6#\"#K<%<$&F'6#\"#T&F'6#\"#R<$&F'6#\"#A&F'6#\"#X<$&F'6#\"# D<$&F'6#\"#Z&F'6#\"#c<%&F'6#\"#><$<$&F'6#\"#G<$&F'6#\"#=<&&F'6#\"#6&F' 6#\"#L&F'6#\"#]<$&F'6#\"#7&F'6#\"#I&F'6#\"#S<%&F'6#F#&F'6#\"#O<$&F'6# \"#[<$&F'6#\"#V&F'6#\"#Q<$<$&F'6#\"\")&F'6#\"#?&F'6#\"#\\<%&F'6#\"#F<' &F'6#\"#@<$<$&F'6#\"#8<&&F'6#\"#5&F'6#\"#H&F'6#\"#N<$&F'6#\"#J<%&F'6# \"#e<$<%&F'6#\"#M<$&F'6#\"\"(&F'6#\"#`<$&F'6#\"#P&F'6#\"#_&F'6#\"#Y<$& F'6#\"#;&F'6#\"#d<$&F'6#\"\"&<$&F'6#\"\"$&F'6#\"#B<$&F'6#\"\"%<$<$&F'6 #\"\"#&F'6#\"#9<%&F'6#\"#:&F'6#\"#<<$<$&F'6#\"#U<$<%&F'6#\"\"'&F'6#\" \"\"&F'6#\"#C&F'6#\"#a&F'6#\"#^&F'6#\"#f&F'6#\"#W<$&F'6#\"#E&F'6#\"#b " }}{PARA 12 "" 1 "" {XPPMATH 20 "6$\"#!)<$<$<$<$<%&%\"ZG6#\"#u<%<$<%& F*6#\"#q<%&F*6#\"#p<%&F*6#F#<$&F*6#\"#>&F*6#\"#T<%&F*6#\"#7&F*6#\"#H&F *6#\"#R<$&F*6#\"#v<$&F*6#\"#o&F*6#\"#G<$&F*6#\"#s<%&F*6#\"#t&F*6#\"#f& F*6#\"#^&F*6#\"#W<$&F*6#\"#i<%<$<%<$&F*6#\"\")<$<$<$<$&F*6#\"#=<%&F*6# \"#k<&&F*6#\"#8<$&F*6#\"#K&F*6#\"#U&F*6#\"#C&F*6#\"#c&F*6#\"#J&F*6#\"# I&F*6#\"#X&F*6#\"#]<$&F*6#\"\"&&F*6#\"#N<$&F*6#\"#w<%<$&F*6#\"#z&F*6# \"#Z<$&F*6#\"#x&F*6#\"#m<$&F*6#\"#O&F*6#\"#e<$&F*6#\"#M<%&F*6#\"#9&F*6 #\"#j<$<$<$&F*6#\"\"%&F*6#\"#y&F*6#\"#@<%&F*6#\"#B&F*6#\"#E&F*6#\"#F<$ &F*6#\"#:&F*6#\"#`<$&F*6#\"#<&F*6#\"#;<$&F*6#\"\"(&F*6#\"#l<$&F*6#\"\" *<$&F*6#\"\"#<$<$<%&F*6#\"#r<$<$&F*6#\"#?<&&F*6#\"\"$&F*6#\"#5&F*6#\"# 6<$<$&F*6#\"#V&F*6#\"#A&F*6#\"#d<$&F*6#\"#h&F*6#\"#Q<$<$&F*6#\"#n<$<$& F*6#\"\"\"&F*6#\"#[<$<$&F*6#\"#g&F*6#\"#Y<$&F*6#\"\"'&F*6#\"#\\&F*6#\" #_&F*6#\"#b&F*6#\"#S&F*6#\"#P&F*6#\"#L&F*6#\"#a&F*6#\"#D" }}{PARA 12 " " 1 "" {XPPMATH 20 "6$\"$+\"<%<$<&<$&%\"ZG6#\"\"$<$<%<$<$<$<$<$<%<$<$& F)6#\"#Z&F)6#\"#_<$<$&F)6#\"#\"*<$<$&F)6#\"#%)&F)6#\"#N&F)6#\"#d&F)6# \"#v<$<%<$<%<$<%<$<$<$<$<$<$&F)6#\"\"&<%<$<$<$<$&F)6#\"#l&F)6#\"#b<$&F )6#\"#:&F)6#\"#;&F)6#\"#'*&F)6#\"#R<$&F)6#\"#$*&F)6#\"#c&F)6#\"#Q<%&F) 6#\"#**&F)6#\"#(*&F)6#\"#Y<$<$<$&F)6#\"\"#<$<%<$<$<%&F)6#\"#s&F)6#\"#L &F)6#\"#H&F)6#\"#F<%<$&F)6#\"\"\"&F)6#\"#=&F)6#\"#O&F)6#\"#e<$&F)6#\"# 8&F)6#\"#`&F)6#\"#)*&F)6#\"#P&F)6#\"#g<%<$&F)6#\"#a&F)6#F#&F)6#\"#k&F) 6#\"#))<$&F)6#\"#o&F)6#\"#C&F)6#\"#B<$<%<$<%&F)6#\"\")<$<$&F)6#\"\"(&F )6#\"#G<$&F)6#\"#y&F)6#\"#V&F)6#\"#i<$&F)6#\"#\")&F)6#\"#x&F)6#\"#E&F) 6#\"#U&F)6#\"#[<$<$&F)6#\"#()&F)6#\"#j&F)6#\"#z&F)6#\"#t&F)6#\"#$)<$&F )6#\"#n&F)6#\"#&)&F)6#\"#9&F)6#\"#f<$<$<$&F)6#\"#!)&F)6#\"#]&F)6#\"#@< $<$<$&F)6#\"\"*<%&F)6#\"\"'&F)6#\"#I&F)6#\"#X&F)6#\"#D&F)6#\"#T&F)6#\" #M&F)6#\"#p&F)6#\"#^&F)6#\"#h&F)6#\"#K<$<$<$<&<$&F)6#\"#')&F)6#\"#\\<$ <$&F)6#\"#r&F)6#\"#J&F)6#\"#5<%&F)6#\"\"%&F)6#\"#?&F)6#\"#&*&F)6#\"#w& F)6#\"#<<$&F)6#\"##*&F)6#\"#%*&F)6#\"#!*&F)6#\"#*)&F)6#\"#>&F)6#\"#m&F )6#\"#u&F)6#\"#q&F)6#\"#7&F)6#\"#A&F)6#\"#W&F)6#\"##)&F)6#\"#6&F)6#\"# S" }}}{PARA 0 "" 0 "" {TEXT -1 30 "The number of objects of size " } {XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 11 " grows fast" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(count(hier,size=j),j=0..40);" } }{PARA 12 "" 1 "" {XPPMATH 20 "6K\"\"!\"\"\"F$\"\"%\"#E\"$O#\"%_F\"&3# R\"'K+m\")7*=G\"\"*Cy8#G\"+cy*)Rp\"-%y#=m')=\".W0-\\th&\"0G(4Kq!z\"=\" 1si[Ugs`j\"3GxDl4(R^Q#\"43E,>ke?5d*\"6'\\5SWgc!z$)3%\"8Cm!p)\\O5aIA&= \"9;%)QZ$>TI6Z4())\";;%3x)\\z_x&)=AyW\"=K;a&yjK')G>k8mP#\"?Gt#ei(>RbzW k5!GK\"\"@KgRKx_j*\\(Hnr3]q(\"B_zNN$o#eDdE,!e8W(o%\"Dkoz0?Q9xZ*4L+Ra+t H\"FciXN:**es=tnhPJ.'ei>\"HC--[(*\\q_L\"oddi/%fwjM\"\"I%=`m=B?\"GGdk:5 =)[kj\\e*\"Ko^8g9E^/Mtrq\"M#**G)f\")p()[)4#G*4w#fy\"\\8[0S&\" O)3&=W1Xj!\\B#p#=sMLx%G_q4kU\"Q3e`O)*o$H0Fu%y4'*)el37W%\\,u7-pY# Hao#z/HUNN(*z!o&)eE#\"Ycq.x'=hEzg)fFqV?r8E9JHu\\0)\\B3#\"ens1!zW!p5e?_ <1e)*y::)o\"=Ln]gN`en>\"gnoN,wob1SU!z2;!o?kjKGh'Q/8!*Q6!35>\"in7h]GK#[ X&>(f8Nmw^aTKgd\\\\,w)[37q.>" }}}{PARA 0 "" 0 "" {TEXT -1 28 "This app ears to be sequence " }{TEXT 273 5 "M3613" }{TEXT -1 8 " of the " } {TEXT 274 33 "Encyclopedia of Integer Sequences" }{TEXT -1 241 " and i t corresponds to \"Schroeder's fourth problem\". When the count is not too large, we can do exhaustive listings. This is made possible by Co mbstruct that is able to build canonical forms and generate elements u nder unique standard forms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "for j to 4 do map(lreduce,allstructs(hier,size=j)) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#&%\"ZG6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#<$&%\"ZG6#\"\"#&F&6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&<$<$&%\"ZG6#\"\"\"&F'6#\"\"$&F'6#\"\"#<$F*<$F-F&<%F-F &F*<$<$F-F*F&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7<<$&%\"ZG6#\"\"\"<%& F&6#\"\"#&F&6#\"\"%&F&6#\"\"$<$F%<$F*<$F-F0<$F0<%F*F-F%<$F%<$F0<$F*F-< $F%<$<$F*F0F-<$F5<$F*F%<$F*<$<$F%F0F-<$F*<%F-F%F0<$F0<$F-F?<%F-F0F?<%F *F%F5<%FBF*F-<$<$F=F%F-<$F*<$F%F5<&F*F-F%F0<$<%F*F%F0F-<$<$F0F?F-<$F=< $F-F%<$FBF:<$F*<$F0FT<%F=F-F%<$F0<$F*FT<$F0<$F%F:<%F%F0F:<$<$FBF*F-<%F *F0FT" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Asymptotic analysis" } }{PARA 0 "" 0 "" {TEXT -1 40 "We get generating function equations by \+ " }{HYPERLNK 17 "combstruct[gfeqns]" 2 "combstruct[gfeqns]" "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "gfeqns(op(2..3,hier),z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$/-%\"ZG6#%\"zGF(/-%\"HGF',*F%\"\"\"- %$expG6#F*F-!\"\"F-F*F1" }}}{PARA 0 "" 0 "" {TEXT -1 4 "And " } {HYPERLNK 17 "combstruct[gfsolve]" 2 "combstruct[gfsolve]" "" }{TEXT -1 50 " attempts different strategies to solve the system" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "gfsolve(op(2..3,hier),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/-%\"ZG6#%\"zGF(/-%\"HGF',(-%)LambertWG6#, $-%$expG6#,&F(#\"\"\"\"\"##!\"\"F7F6F8F9F(F5F8F6" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 22 "The solution involves " }{HYPERLNK 17 "Lambert's W function" 2 "LambertW" "" }{TEXT -1 63 " that is known to Maple: by d efinition, this is the solution of" }}{PARA 264 "" 0 "" {XPPEDIT 18 0 "W(z)*exp(W(z))=z" "/*&-%\"WG6#%\"zG\"\"\"-%$expG6#-F%6#F'F(F'" } {TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "H_z:=subs(\",H(z)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$H_zG,(-%)LambertWG6#,$-%$expG6 #,&%\"zG#\"\"\"\"\"##!\"\"F1F0F2F3F.F/F2F0" }}}{PARA 0 "" 0 "" {TEXT -1 73 "Objects being labelled, this is an exponential generating funct ion (EGF)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "H_ztayl:=serie s(H_z,z=0,20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(H_ztaylG+K%\"zG\" \"\"\"\"\"#F'\"\"#\"\"##F*\"\"$\"\"$#\"#8\"#7\"\"%#\"#f\"#I\"\"&#\"$s \"\"#X\"\"'#\"%,\\\"$I'\"\"(#\"&8.\"F=\"\")#\"'\"f+%\"&S8\"\"\"*#\"(2o \"))\"'+M6\"#5#\")w*3r#\"'Df:\"#6#\"+`X&RZ\"\"(+Au$\"#7#\",B#Rb)Q%\")+ '['[\"#8#\"-8W$*>,r\"*+-aS$\"#9#\"/Jn<@'4C\"\"++:0aD\"#:#\"/jRtIW$e$\" ++![M9$\"#;#\"1<[U\"z*pY$*\"-+S+^tM\"#<#\"3xOEBO)yi*>\".+g.fh7$\"#=#\" 3^P@/Ad*p&p\".+!o;+pX\"#>-%\"OG6#F'\"#?" }}}{PARA 0 "" 0 "" {TEXT -1 79 " As usual, we also obtain the corresponding ordinary generating fu nctions by a " }{HYPERLNK 17 "Laplace transform" 2 "inttrans[laplace] " "" }{TEXT -1 32 " applied to the series expansion" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "series(subs(w=1/w,w*inttrans[laplace](H_zta yl,z,w)),w,20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+I%\"wG\"\"\"\"\"\" F%\"\"#\"\"%\"\"$\"#E\"\"%\"$O#\"\"&\"%_F\"\"'\"&3#R\"\"(\"'K+m\"\")\" )7*=G\"\"\"*\"*Cy8#G\"#5\"+cy*)Rp\"#6\"-%y#=m')=\"#7\".W0-\\th&\"#8\"0 G(4Kq!z\"=\"#9\"1si[Ugs`j\"#:\"3GxDl4(R^Q#\"#;\"43E,>ke?5d*\"#<\"6'\\5 SWgc!z$)3%\"#=-%\"OG6#F%\"#>" }}}{PARA 0 "" 0 "" {TEXT -1 68 "The resu lt is then directly comparable to the counting coefficients:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(count(hier,size=j),j=1.. 18);" }}{PARA 12 "" 1 "" {XPPMATH 20 "64\"\"\"F#\"\"%\"#E\"$O#\"%_F\"& 3#R\"'K+m\")7*=G\"\"*Cy8#G\"+cy*)Rp\"-%y#=m')=\".W0-\\th&\"0G(4Kq!z\"= \"1si[Ugs`j\"3GxDl4(R^Q#\"43E,>ke?5d*\"6'\\5SWgc!z$)3%" }}}{PARA 0 "" 0 "" {TEXT -1 155 "In order to analyse the number of hierarchies, we m ust find the dominant singularity of their generating function. A plot detects a vertical slope near 0.4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "plot(H_z,z=0..1);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%' CURVESG6$797$\"\"!F(7$$\"+;arz@!#6$\"*4(=/A!#57$$\"+XTFwSF,$\"*z#>kTF/ 7$$\"+\"z_\"4iF,$\"*`'p>kF/7$$\"+S&phN)F,$\"*&eV]()F/7$$\"+*=)H\\5F/$ \"+L+o86F/7$$\"+[!3uC\"F/$\"+!4g:M\"F/7$$\"+J$RDX\"F/$\"+B22&e\"F/7$$ \"+)R'ok;F/$\"+kQ2Y=F/7$$\"+1J:w=F/$\"+Y1!p6#F/7$$\"+3En$4#F/$\"+;ZY3C F/7$$\"+/RE&G#F/$\"+VW-VF/7$$\"+347TLF/$\"+*3Wnb%F/7$$ \"+rxdOMF/$\"+nV.$z%F/7$$\"+LY.KNF/$\"+go^b]F/7$$\"+\"o7Tv$F/$\"++Hzpe F/7$%%FAILGF_r-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$%\"zG%!G -%%VIEWG6$;F($\"\"\"F(%(DEFAULTG" 2 264 264 264 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 137 "Here is a cute wa y to get the singularity \"automatically\". We express that the funct ion ceases to be differentiable at its singularity. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "diff(H_z,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%)LambertWG6#,$-%$expG6#,&%\"zG#\"\"\"\"\"##!\"\"F 0F/F1F/,&F/F/F%F/F2F1F.F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "rho:=solve(denom(\")=0); evalf(rho, 30);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoG,&!\"\"\"\"\"-%#lnG6#\" \"#F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\">#HCkW$)=1*)>6O%H'Q!#H" }} }{PARA 0 "" 0 "" {TEXT -1 106 "Next, we know that the singular expansi on determines the asymptotic form of coefficients. Thus, we look at" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "H_s:=subs(z=rho*(1-Delta^2) ,H_z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$H_sG,(-%)LambertWG6#,$-%$ expG6#,&*&,&!\"\"\"\"\"-%#lnG6#\"\"#F5F1,&F1F1*$%&DeltaGF5F0F1#F1F5#F0 F5F1F:F0F.F9F:F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "H_sing: =map(simplify,series(H_s,Delta=0,5));Delta=sqrt(``(1-z/rho));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'H_singG+-%&DeltaG-%#lnG6#\"\"#\"\"! ,$*$,&!\"\"\"\"\"F'F*#F0F*F/\"\"\",&#F0\"\"'F0F'#F/\"\"$\"\"#,$*$F.#F7 F*#F/\"#O\"\"$-%\"OG6#F0\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&De ltaG*$-%!G6#,&\"\"\"F**&%\"zGF*,&!\"\"F*-%#lnG6#\"\"#F2F.F.#F*F2" }}} {PARA 0 "" 0 "" {TEXT -1 136 "With this, we can get an asymptotic expa nsion for coefficients to any order, which is an interesting fact per \+ se. Here is the first one:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "H_n_asympt:=n!*asympt(coeff(H_sing,Delta,1)*rho^(-n)*subs(\{cos(P i*n)=1,O=0\},simplify(asympt(binomial(1/2,n),n,2))),n);\nevalf(\",20); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+H_n_asymptG,$*,-%*factorialG6#% \"nG\"\"\",&!\"\"F+-%#