æáó /=”ôø  $¤N*OņP!Q¨‚Q¼Q3 Sü1V<WV7ªYµXZ -[6‰\uÕ\?ý\Ë ˜][å] ^XÌ^ýà^§_} Ü_ `M§bD |cód *dÜžd“†i¢jÏÎjyBkïël« ÔmOoË}oÂÊo§yp"Ÿp¼9qÌÁq8•rÕ s òs¥ t!v Iv2kwçîx2:y uy 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Head ing 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 8 2 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 "Bullet Item" 0 15 version2P Ÿ/{VERSION 3 0 "DEC ALPHA UNIX" "3.0" } {USTYLETAB {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 2 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 " " -1 256 "" 1 10 0 0 0 0 1 0 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 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 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 "" 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 }{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 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -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 1 0 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 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 295 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 296 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 300 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 301 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 302 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 303 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 304 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 305 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 306 "" 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 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 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 15 2 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 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 18 "" 0 "" {TEXT -1 19 "The algolib Library" }} {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 256 37 "Comb inatorics meets computer algebra!" }}{PARA 258 "" 0 "" {TEXT -1 0 "" } }{PARA 19 "" 0 "" {TEXT -1 0 "" }{TEXT 257 18 "Algorithms Project" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Th e " }{TEXT 258 7 "algolib" }{TEXT -1 21 " library consists of " } {TEXT 259 5 "Maple" }{TEXT -1 28 " packages developped by the " } {TEXT 274 18 "Algorithms Project" }{TEXT -1 550 ", a research group at INRIA Rocquencourt (France). Our packages are primarily intended for the manipulation of combinatorial structures (specification, generati on, enumeration, computation of generating functions), for their asymp totic analysis, and the applications to the automatic complexity analy sis of algorithms, but the library also contains packages for the mani pulation of linear differential and difference operators, Groebner bas is calculations, and the symbolic summation and integration of special functions and combinatorial sequences." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 "The algolib Pack ages" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Here is the list of the " }{TEXT 296 7 "algolib" }{TEXT -1 10 " packages:" }}}{EXCHG {PARA 15 " " 0 "" {HYPERLNK 17 "combstruct." 2 "combstruct" "" }{TEXT -1 74 " Co mbinatorial structures package (version 4.0, December 12, 2000). The \+ " }{TEXT 264 10 "combstruct" }{TEXT -1 228 " package is used to define and manipulate a wide range of combinatorial structures. Structures \+ can be counted and generated uniformly at random, and in some cases it is possible to generate all the structures of a given size. " } {TEXT 260 112 "[Written by Eithne Murray and Paul Zimmermann, with con tributions by Marni Mishna. Comments and bug reports to " }{TEXT 282 19 "combstruct@inria.fr" }{TEXT 283 2 ".]" }}{PARA 15 "" 0 "" {HYPERLNK 17 "encyclopedia." 2 "encyclopedia" "" }{TEXT -1 118 " An e ncyclopedia of combinatorial structures (July 22, 2000). An on-line v ersion of it is also available at the URL " }{TEXT 301 44 "http://algo .inria.fr/encyclopedia/index.html" }{TEXT -1 3 ". " }{TEXT 261 111 "[ Written by St\351phanie Petit, with contributions by Bruno Salvy and M ich\350le Soria. Comments and bug reports to " }{TEXT 284 29 "encyclo pedia@pommard.inria.fr" }{TEXT 285 2 ".]" }{TEXT -1 0 "" }}{PARA 15 " " 0 "" {HYPERLNK 17 "gdev." 2 "gdev" "" }{TEXT -1 140 " A facility fo r more general series expansions and limits (last update May 9, 2001). Uses a different model fo asymptotic expansions than " }{TEXT 268 5 "Maple" }{TEXT -1 3 "'s " }{TEXT 269 6 "asympt" }{TEXT -1 5 " and " } {TEXT 270 6 "series" }{TEXT -1 260 " commands. Includes the equivalen t function mentioned in the survey article (see below) for the asympto tic expansions of coefficients of generating functions. It does asymp totic expansion of Taylor coefficients, useful in the study of generat ing functions. " }{TEXT 267 25 "[Written by Bruno Salvy.]" }}{PARA 15 "" 0 "" {HYPERLNK 17 "gfun." 2 "gfun" "" }{TEXT -1 67 " Generating functions package (version 2.63, June 23, 1999). The " }{TEXT 265 4 "gfun" }{TEXT -1 78 " package is used for the manipulation and discove ry of generating functions. " }{TEXT 262 111 "[Written by Bruno Salvy and Paul Zimmermann, with contributions by Eithne Murray. Comments a nd bug reports to " }{TEXT 286 13 "gfun@inria.fr" }{TEXT 287 2 ".]" }} {PARA 15 "" 0 "" {HYPERLNK 17 "Groebner." 2 "Groebner" "" }{TEXT -1 158 " Groebner bases package (version 3.0, January 3, 2000). Impleme nts Groebner basis methods in commutative polynomial algebras and skew polynomial algebras. " }{TEXT 271 58 "[Written by Fr\351d\351ric Chy zak. Comments and bug reports to " }{TEXT 289 24 "frederic.chyzak@inr ia.fr" }{TEXT 288 2 ".]" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Holonomy." 2 "Holonomy" "" }{TEXT -1 48 " Package for the manipulation of holono mic and " }{XPPEDIT 18 0 "d;" "6#%\"dG" }{TEXT -1 188 "-finite functio ns (version 3.0, January 3, 2000). Deals with functions and sequences that are implicitly defined as solutions of systems of linear differe ntial and difference equations. " }{TEXT 272 58 "[Written by Fr\351d \351ric Chyzak. Comments and bug reports to " }{TEXT 291 24 "frederic .chyzak@inria.fr" }{TEXT 290 2 ".]" }}{PARA 15 "" 0 "" {HYPERLNK 17 "M gfun." 2 "Mgfun" "" }{TEXT -1 81 " Multivariate generating functions \+ package (version 3.0, January 3, 2000). The " }{TEXT 266 5 "Mgfun" } {TEXT -1 246 " package is intended for calculations with multivariate \+ generating functions, in particular for their symbolic summation and i ntegration, and for the proof of special function and combinatorial id entities. It is a user-oriented interface to the " }{HYPERLNK 17 "Hol onomy" 2 "Holonomy" "" }{TEXT -1 34 " package. More information about " }{TEXT 302 5 "Mgfun" }{TEXT -1 24 " to be found at the URL " } {TEXT 303 38 "http://algo.inria.fr/chyzak/mgfun.html" }{TEXT -1 3 ". \+ " }{TEXT 263 93 "[Written by Fr\351d\351ric Chyzak, with contributions by Cyril Germa. Comments and bug reports to " }{TEXT 293 24 "frederi c.chyzak@inria.fr" }{TEXT 292 2 ".]" }}{PARA 15 "" 0 "" {HYPERLNK 17 " Ore_algebra." 2 "Ore_algebra" "" }{TEXT -1 146 " Ore algebras package (version 3.0, January 3, 2000). A package for the manipulation of li near operators and skew polynomials (Ore operators). " }{TEXT 273 58 "[Written by Fr\351d\351ric Chyzak. Comments and bug reports to " } {TEXT 295 24 "frederic.chyzak@inria.fr" }{TEXT 294 2 ".]" }}{PARA 15 " " 0 "" {HYPERLNK 17 "regexpcount." 2 "regexpcount" "" }{TEXT -1 467 " \+ Counting matches of regular expressions (version 1.3, November 29, 20 00). A package for general manipulations of regular expressions and ( marked) automata, with application to computing the probability distri butions of motifs occurrences in a random text and of waiting times fo r first matches. Available models of random texts are: uniform and no n-uniform Bernoulli models, and Markov model. Approximate matching wi th bounded number of errors is also handled. " }{TEXT 304 58 "[Writte n by Pierre Nicod\350me. Comments and bug reports to " }{TEXT 305 24 "pierre.nicodeme@inria.fr" }{TEXT 306 2 ".]" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 50 "Documentation, Demos, and Related Research Papers." }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Beside usual " }{TEXT 297 5 "Maple " }{TEXT -1 21 " on-line help pages, " }{HYPERLNK 17 "demonstration pa ges" 2 "autocomb" "" }{TEXT -1 5 " for " }{TEXT 280 4 "gfun" }{TEXT -1 2 ", " }{TEXT 281 10 "combstruct" }{TEXT -1 91 ", and other combina torial packages are available in the form of combinatorial case studie s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 281 "Th is software is part of an overall framework to study decomposable comb inatorial structures and their generating functions. A survey article by covers this aspect or our research (Computer Algebra Libraries for Combinatorial Structures, Philippe Flajolet and Bruno Salvy (1995), \+ " }{TEXT 279 31 "Journal of Symbolic Computation" }{TEXT -1 125 ", 20: 5-6, p. 653-671). A list of more research papers related to theoretic al aspects relevant to this software can be found " }{HYPERLNK 17 "her e" 2 "algolib,references" "" }{TEXT -1 1 "." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "Updating your algolib Library" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 70 "Parts of the library have been introduced in various pa st releases of " }{TEXT 278 5 "Maple" }{TEXT -1 20 ", specifically, th e " }{TEXT 298 10 "combstruct" }{TEXT -1 2 ", " }{TEXT 299 11 "Ore_alg ebra" }{TEXT -1 6 ", and " }{TEXT 300 8 "Groebner" }{TEXT -1 131 " pac kages. The present distribution contains more recent releases of thes e packages, which fix bugs and/or add new functionality. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "The most recent ve rsions of the library will be available:" }}{PARA 0 "" 0 "" {TEXT -1 35 " + by anonymous ftp on the site " }{TEXT 275 12 "ftp.inria.fr" }{TEXT -1 12 ", directory " }{TEXT 277 29 "INRIA/Projects/algo/program s/" }{TEXT -1 1 ";" }}{PARA 0 "" 0 "" {TEXT -1 28 " + on the web at the URL " }{TEXT 276 31 "http://algo.inria.fr/libraries/" }{TEXT -1 1 "." }}}}}{MARK "1 0 6" 550 }{VIEWOPTS 1 1 0 3 2 1804 } }{TEXT -1 1 "." }}}}{SECT 1 {PARA 3 "" olbg†Pold¿ ëlolderjolexprtodiffeqkolf Õ\ëloliàolic ï"oll "Îjollard Bk!vollect½8†PÎjolleuz"ollllÕ\ollow‡/olon ë !Qolonomic-BolqbkŠ olsÕ\ oltodiffeq¿ olutJ@olvÕ\olvablCNolvableg‡/olve" A†P‰\olwëloly¿ ½8ü; ^dolyg ¿ †Polynom‚Q olynomial¿ .FBk!vom; àŠ ¿ "Õ\Ì^Ü_ÎjBkëlo:yomaëlomatŠ omatonombÕ\ ombinaison½8 ombstruct+ {"w" ALD¼Qý\§bël»MÒdecalphaunixmaplinputcourimathtimehyperlinkcommoutputourihelpheadnormallistitembulletfunctorealgebradiffcreatlineardifferentialoperatorusagcallsequencalgebraglgigngparameterdgxglistcommgagindeterminatvariablnamesynopsidescriptdeclarreturntablwillusedotherfunctionpackagweylnonutatpolynomialgfruledfollowcommutatidentitpaircommutnotespecialcasefurtherinformatassignedsuchassigndxdxgdenoterentialassociatbasedifferentiatwithrespectformassignedbothommutanyparametotthoughtheyrepresentstandardproducteverfunctiodealelementusesllappearrightcorrespondmonomialbgfcanthereforprinteitherbgagfsumperformsimpusingwhilskewalgebraexamplbelowalsopossibloptionavailablcontrolgroundringactionobjectdeclaratpartonlyaftercommandalgebralwayaccesslongwitnadydxgfxgfdygfygfequivalfirstsyntaxmoreconveninumerouerscomseealsolgebraproducrefofafa:yfofafj‡/fofef¿ fofkf½fofnëlfofo Öµ%fofofhëlfofqÖfofrji†PfogÕ\fojÕ\fol †P¼QfollÕ\folloW5òsfollowûÝ3àiŠ ¿ £ Æ:?¯KÍ "µ%'3(Ô+‡/ƒ4½4W5R67½8ü;$@J@ àALD.F»M*O†P!Q‚Q¼Q SªY‰\Õ\ ý\ ^à^Ü_djÎjBkël oÊoyp9qÁq•ròstkw:yfollowi9J@Êo:yfollowin¿ eal ë ¿ ealso{eangÎjeans Îj:year¿ Õ\•rearancÜ_earch Œkearl¿ KàAearli†P¼QÎj•r!v:yƒ4½4R67½8S:ˆ; ü; <¥=˜@àA.F»I»M *O ‚Q¼QªY‰\Õ\ ^à^Ü_*d†ioÊo9qòs!vqccÕ\qcesgëlqcf¿ "Õ\ëlqcgnfëlqcj†Pqck†PqcrëlqcsW5qczhÎjqdŠ ¿ "Õ\ël9qqdad¿ qdbÕ\qdblbëlqdeltaR6_9qqdfŠ ¿ Õ\ëlqdfduÕ\qdfqëlqdiff _9qqdilat3(ˆ;å]_9qqdjdÎjqdqtdëlqdwÕ\qdxëlqe ¿ "†PÕ\ëlqea¿ qeas¿ qeb¿ qeblk†PqecëlÕ\xmmm"xmmmmmmÕ\xmt¿ xmtf¿ xnŠ "ëluah¿ uaio†PuajÕ\uapbomfëluare!vuarfÕ\uation# ¿ $@àA¶J‚QÕ\Ü_ub ¿ èÕ\ëlubctëlubf"ubqf¿ ubsŠ J@¼QBkël:yubsjp¿ ubstitutW5ubtîxubtractÎjubuxefubvëlubydëluc àÕ\ëluccess¼Quce ó=¼QucedŸpuceuW5uch¿ ucket¼QucobkÕ\ucosnÕ\ucp"uct7Æ¿"ƒ4R6 A*O§bëlÔmÊoIvîxuction9ucturIvud ¿ ".FÕ\ëludaÕ\udeÎjudfÕ\udiÕ\udjazÕ\udln¿ udo"udoy¿ ue¿ 9Õ\ëlîxuef"uegÞ5uegcëlÕ\!v decalphaunixmathtimehyperlinkcommnormalheadbulletitemtitlauthorstudautomaticcombinatoriccombinatoricmeetcomputeralgebraalgorithmprojecthereserinotedescribinteractbetweencombinatorialanalysidiscretmathematictheydiscusexploratusingmaplsystemsymboliccomputconjunctwithpackaglikecombstructgfunmgfunappearvolumvolumeiiavailablwebsitpresworksheetlistcasesortapplicatdomainalsopointadditionaltutorialintroductseveraltheshaveproducearlireleasalgolibupdatyoumaythusgetifferoutputreproducthemtroducthestructurgrammarlightlllearnaboutbasicspecificathowcountsequencusepredefinsubsetpermutatcombinatetcsamplstructsimplcollectexamplshowgeneratrandomtreeinvestigatthdistributheightsimulatenumeratfunctionalgraphlcoholnecklacexpressfunctionstartversbecompossiblfunctequationsolvsomenewcallallstructexhaustaddedoreinearoperatorskewpolynomialintrointegratsummatnaturaldescripttZ£ ¾ decalphaunixmaplinputcourimathtimehyperlinkcommoutputcouriernormalheadtitlauthorasymptoticstirlingnumbersecondkindbrunosalvjohnshackeljanuarweshowhowcomputalgebracanusedautomaticalasymptoticbellaveragvaluvariancnumbpartpartitworksheetdetailcomputationinvolvsectourarticlsymbolicsymptoticmultiserinversfunctionavailableinriaresearchreportstartpointalltheseexpansbivariatgeneratfunctexpfgexpgugxgfserixgcffcfogcoefficikgngfsgngdividfactorialgthesnumberscountsetionswithfastgrowthatinfinitlikemanyrelatbehavioutaylorcoefficientsaddlmethomethodreliusingintegralalongcloscontourhecomplexplanchosensuchwaypassthroughspecialpointcallconcentratneighborhoodgoodestimatobtainlocalhowevapplicatsadddefinimplicitequatonlyxpansavailablintroductechnicaldifficultmanipulatexplorlgorithmapplcircumstancfirstloadexperimentalimplementatreadexporalcomlibasymptinvmpl *O*ibmintellinuxmaplinputcourimathtimehyperlinkcommoutputourihelpheadnormaltextwarningerrorlistitembulletfunctorealgebraskewdeclarsagecallsequenctglgngoptionparameterigtypecommutatindeterminatwhoslengthdetermincorrespondoptionaldescribbelowsynopsidescriptlgebrareturntablwillusedotherfunctionpackagrepreslinearoperatorapplsuchlineardifferentiatrecurrencpolynomialsrgruledfollowpseudidentitanypolynomialpgxgsigmadeltadgpgfigmagdeltagpairjgutessigmasigmagendomorphismadditmoreoversatisfleibnitzruleqgfnoteweylspecialcaseobtainalldifferentiatsealgebrafurthinformatinvolvnonassignnameeachconsistpseudodifferentiastringrepresenthederivateithpredefinuserdefinpossiblalgebrathoughtheywithstandardcommutativeproducteverdealelementusesformappearrightcorrespondingmonomialagbgfmeantcantherefordisplayitherbgagfsumperformsimpusingwhilexamplavailablecontrolgroõqlfëlqlkz¿ qll ¿ Îjqlll"qlllf"qllllÕ\qlllllÕ\qloxznëlqlpÕ\qlsëlqlvÕ\qlwÎjqlyjcÎjqlyx¿ qlyzÕ\qlzÕ\qlzumtàAqm¿ "W5àAÕ\ëlqmcëlqmf"qmfmëlqmgfëlqmkqëlqml¿ qmmm"qmmmrd"qmn¿ qmslÕ\qmtf¿ qmxaÎjqmy¿ qmz ¿ ëlqn'àŠ ¿ "3(Õ\Îjëlîx"3(ú+ ,V1½4W5Þ5R67½8S:Â;<¥=ó=$@J@˜@ AàA-B”E.F»I¶J»MCN*O!Q‚Q¼Q%WVXZÕ\1å] ^Ì^à^_§bd†iÎjBk ëloÊoŸpÁq•ròs)!vkw:yewfduÕ\¼QWVXZ‰\Õ\å] ^Ì^_dÎjBkëloÊo9qÁq•ròs!vkw:ydecalphaunixmaplinputcourimathtimehyperlinkcommoutputnormalextheadwarnerrorplotenumerationplanarconfiguratcomputationalgeometrphilippflajoletversjanuarconsidngpointcircldefinconvexpolygonenumeratgeometriccansuperimposthesdignifihistoreulersolvproblemcountnumbpossibltriangulatgonwhilinventonceptgeneratfunctparticularcombinatorialenumationsincmanyhaveenumerateecomtetadvanccombinatoricdiscussworkprouhetjordanguytillrecentanynewresultbasicalresearchpapersomeapplicatbelowrequirseveralpagesrecurrencmanipulatreportwetakeinspiratfrommemonoyseptembshowsuchresultsderivautomaticalusingsystempackagcombstructgfunworsheetmeantsimplintroductspecificatwellbasicexperimentalcomputatmayaccompanthemfindwayscuttupintotrianglmeannonintersectdiagonalhereexamploctogonicosagoncurvesgrqrqfknkmfbygukfjfgpofufjnfgnfufufofiofjfiofiofofjfuaxesstylegnonegcolourgrgbgfgcKmuozëlmurra P •rmus ,Õ\:ymust«J3i.ç CÆ:½k"Ô+ú+ ,Ð0½4X9<ó=$@8@¤N†P¼Q S‰\ý\˜]Ü_ )`§bždÎjÔm9q sòskwîx:ymustb”EmutatÊomutation Îj:ymutualjmv¿ Õ\:ymvbÕ\mveàAmvfÕ\mvgrfÕ\mvhÕ\mvjÕ\mvlxëlmvpëlmvrÕ\mvrkÕ\mvrkfÕ\mvrkpfÕ\mwëlmwaÎjmweightïmwfgŠ mwvuÕ\mwxisaŠ mx ¿ ëlmxnŠ mxngŠ mxpn¿ mxwqgmtÕ\mxymÕ\my¿ Æ3(Õ\ëlmydifÆmydiffÆmyfëlmyihÕ\myjël:J@-BÕ\Bkëlòstur †Pmod Bk mode ¼Qëlmodel?YàŠ ¿ P 'W5˜@¼QÕ\ÎjBk ël!vîx:ymodell:ymoderateàmodgBkmodiëlmodif¼QÎjBkëlmodifi à¿ $@¼QÎjBkël modificat Îj:ymodifier¼QmodifyX9¤Nëlmodul''ë ? 3( ,‡/¶J ^Êokwmodular ^moduliŠ moduloÍ <å]:ymodulu Š Õ\ždmoeëlmol!vmolecul àŠ W5¼Qîxmom”E‰\ëlmomen  Aòsmoment;7£ Æ:" $@ A”E †P¼Q‰\Ü_òs !vmomentg:' AÜ_Îj:y momentsolvÜ_mon R6*Omonic ˆ;.F monoideal ‡/½8monomŠ ^!vP ùdecalphaunixmathtimehyperlinkcommnormalheadbulletitemtitlauthoralgoliblibrarcombinatoricmeetcomputalgebraalgorithmprojectthconsistmaplpackagdeveloppresearchgroupatinriarocquencourtfrancourprimariintendmanipulatcombinatorialstructurspecificatgeneratienumeratcomputatgeneratfunctiontheirasymptoticanalysiapplicatautomaticcomplexitanalsisbutalsocontainmanipulatlineardifferentialdifferencoperatorgroebnbascalculatsymbolicsummatintegratspecialsequencpackagesherelistcombstructcombinatorialversdecembuseddefinwiderangcancountuniformrandomsomecasepossiblallgivensizewritteneithnmurrapaulzimmermanwithcontributmarnimishnacommentbugreportfrencyclopediancyclopediajulylineersionavailablurlhttpalgoindexhtmlstphanipetitcontributbrunosalvichlesoriaencyclopediapommardgdevfacilitfomoregeneralseriexpanslimitlastupdatmayusesdiffermodelasymptoticasymptcommandincludequivalenfunctmentionsurvearticlbelowasymptoticcoefficientdoestaylorusefulstudinggfunjunediscovryndbasejanuarimplemntsbasimethodcommutatpolynomialskewricchyzakfredericchyzakinriaholonomholonomicdgfinitfunctionsdealimplicitsolutionsystemntialequationmgfunmultivariatparticularntegratproofidentituserorientinterfacholonominformataboutfoundcyrilgermafrederiorelinearregexpcountmatchregularexpressnovembmarkautomataprobabilitdistributionmotifoccurrenctextwaitfirstnobernoullimarkovapproximatwiboundnumberrorhandlwrittpierrnicodmenicodemdocumentatdemorelatpaperbesidusualhelppagedemonstratpagesautocombothercombinatorialformstudisoftwarpartoveralframeworkdecomposablinatorialcoveraspectphilippflajoletjournaltheoreticalrelevantherreferencyourhaveintroducvarioureleasspecificalalgebrapackagepresdistributrecentthesfixbugsaddnewfunctionalitmostversionwillanonymouftpsitedirectorprogramwebonsurvearticlbelowasymptoticcoefficientdoestaylorusefulstudinggfunjunediscovryndbasejanuarimplemntsbasimethodcommutatpolynomialskewricchyzakfredericchyzakinriaholonomholonomicdgfinitfunctionsdealimplicitsolutionsystemntialequationmgfunmultivariatparticularntegratproofidentituserorientinterfacholonominformataboutfoundcyrilgermafrederiorelinearregexpcountmatchregularexpressnovembmarkautomataprobabilitdistributionmotifoccurrenctextwaitfirstnobernoullimarkovapproximatwiboundnumberrorhandlwrittpierrnicodmenicodemdocumentatdemorelatpaperbesidusualhelppagedemonstratpagesautocombothercombinatorialformstudisoftwarpartoveralframeworkdecomposablinatorialcoveraspectphilippflajoletjournaltheoreticalrelevantherreferencyourhaveintroducvarioureleasspecificalalgebrapackagepresdistributrecentthesfixbugsaddnewfunctionalitmostversionwillanonymouftpsitedirectorp*OK#{VERSION 3 0 "IBM INTEL LINUX" "3.0" } {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 2 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 23 "C ourier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "Help Heading" -1 26 " " 1 14 0 0 0 0 0 1 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 Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "War ning" 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 ç£ "_{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 " " 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 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "courier" 0 1 0 0 0 0 0 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 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 }{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 1 0 0 0 0 0 0 0 0 }Îj$`{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 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 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 } {CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 1 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 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 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 }{CSTYL%•r±5{VERSION 3 0 "DEC ALPHA UNIX" "3.0" } {USTYLETAB {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 "" -1 256 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 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 }{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 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 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 1 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 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 1 0 0 0 0 0! 0 0 } {CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" "-1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 292 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 295 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 296 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 300 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 301 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 302 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 303 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 304 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 305 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 306 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 307 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 308 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 309 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 310 "" 0 1 0 0 #0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 311 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 312 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 313 "" 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 8 4 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 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 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 15 2 }{PSTYLE "Title " 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 46 "Research Papers Related t o the algolib Library" }}{PARA 19 "" 0 "" {TEXT -1 29 "(sorted by rele vant packages)" }}{PARA 0 "" 0 "" {$TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "combstruct" }}{EXCHG {PARA 15 "" 0 "" {TEXT 256 64 " A calculus for the random generation of combinatorial structures" } {TEXT -1 59 ", by P. Flajolet, P. Zimmermann, and B. V. Cutsem, (1994) , " }{TEXT 267 28 "Theoretical Computer Science" }{TEXT -1 78 ", vol. \+ 135. Gives the theoretical background of the labelled structures case ." }}{PARA 15 "" 0 "" {TEXT 304 52 "Attribute grammars and automatic c omplexity analysis" }{TEXT -1 57 ", by M. Mishna, (2000), INRIA Resear ch Report, vol. 4021." }}{PARA 15 "" 0 "" {TEXT 307 48 "Attribute gram mars are useful for combinatorics!" }{TEXT -1 42 ", by M. P. Delest an d J. M Fedou, (1992), " }{TEXT 308 28 "Theoretical Computer Science" } {TEXT -1 20 ", vol. 98, p. 65-76." }}{PARA 15 "" 0 "" {TEXT 309 45 "Au tomatic average-case analysis of algorithms" }{TEXT -1 55 ", by P. Fla jolet, B. Salvy, and P. Zimmermann, (1991), " }{TEXT 310 38 "Theoretic al Computer Science, Series A" }{TEXT -1 16 ", vol. 79, n\260 %1." }} {PARA 15 "" 0 "" {TEXT 265 69 "Ga\357a: a package for the random gener ation of combinatorial structures" }{TEXT -1 30 ", by Paul Zimmermann, (1994), " }{TEXT 266 26 "Maple Technical Newsletter" }{TEXT -1 44 ". \+ Ga\357a is an earlier version of combstruct." }}{PARA 15 "" 0 "" {TEXT 311 39 "Lambda-Upsilon-Omega, the 1989 cookbook" }{TEXT -1 110 " , by P. Flajolet, B. Salvy and P. Zimmermann, (1989), INRIA Research R eport, vol. 1073. Lamba-Upsilon-Omega, " }{XPPEDIT 18 0 "Lambda" "6#% 'LambdaG" }{TEXT -1 1 "-" }{XPPEDIT 18 0 "Upsilon" "6#%(UpsilonG" } {TEXT -1 2 "- " }{XPPEDIT 18 0 "Omega" "6#%&OmegaG" }{TEXT -1 59 " or \+ LUO for short, is a previous incarnation of combstruct." }}{PARA 15 " " 0 "" {TEXT 312 54 "Lambda-Upsilon-Omega: an assistant algorithms ana lyzer" }{TEXT -1 59 ", by P. Flajolet, B. Salvy, and P. Zimmermann, (1 989). In: " }{TEXT 313 64 "Applied Algebra, Algebraic Algorithms and E rror-Correcting Codes" }{TEXT -1 64 ", T. Mora (editor), Lecture Notes in Computer Science, v&ol. 357." }}{PARA 15 "" 0 "" {TEXT 306 37 "Obje ct grammars and random generation" }{TEXT -1 39 ", by I. Dutour and J. M Fedou, (1998), " }{TEXT 305 53 "Discrete Mathematics and Theoretica l Computer Science" }{TEXT -1 19 ", vol. 2, p. 49-63." }}{PARA 15 "" 0 "" {TEXT 257 56 "Random generation of unlabelled combinatorial struc tures" }{TEXT -1 79 ", by Eithne Murray, (1993), summary of a seminar \+ talk by Paul Zimmermann. In: " }{TEXT 258 29 "Algorithms Seminar, 199 3-1994" }{TEXT -1 68 ", by Bruno Salvy (editor), (1994), INRIA Researc h Report, vol. 2381." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 4 "gdev" }} {EXCHG {PARA 15 "" 0 "" {TEXT 268 43 "Examples of automatic asymptotic expansions" }{TEXT -1 26 ", by Bruno Salvy, (1991), " }{TEXT 259 15 " SIGSAM Bulletin" }{TEXT -1 16 ", vol. 25, n\260 2." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 4 "gfun" }}{EXCHG {PARA 15 "" 0 "" {TEXT 269 96 "Gf un: a Maple package for the manipulation of generating and holonomic f unctions in one variable" }{TEXT -1 41 ", by B. 'Salvy and P. Zimmerman n, (1994), " }{TEXT 260 41 "ACM Transactions on Mathematical Software " }{TEXT -1 16 ", vol. 20, n\260 2." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 "Ore_algebra, Groebner, Holonomy, Mgfun" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Publications" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "The following are publications that describe the theoretical as pects implemented in the packages." }}}{EXCHG {PARA 15 "" 0 "" {TEXT -1 0 "" }{TEXT 261 36 "Fonctions holonomes en calcul formel" }{TEXT -1 108 ", by Fr\351d\351ric Chyzak, (1998), \"Th\350se universitaire\" n\260\240TU\2400531, INRIA. Defended on May\24027, 1998. 227\240page s." }}{PARA 15 "" 0 "" {TEXT 270 58 "Gr\366bner Bases, Symbolic Summat ion and Symbolic Integration" }{TEXT -1 77 ", by Fr\351d\351ric Chyzak , (1998). In: Buchberger, B. and Winkler, F., (editors), " }{TEXT 262 86 "Gr\366bner Bases and Applications (Proc. of the Conference ``33 \240Years of Gr\366bner Bases'')" }{TEXT -1 167 ", Cambridge Universit y Press (London Ma(thematical Society Lecture Notes Series, vol.\240251 ), p.\24032-60. Preliminary version available as: Research Report n \260 3297, INRIA." }}{PARA 15 "" 0 "" {TEXT 271 74 "An Extension of Ze ilberger's Fast Algorithm to General Holonomic Functions" }{TEXT -1 34 ", by Fr\351d\351ric Chyzak, (1997). In: " }{TEXT 263 63 "Formal Po wer Series and Algebraic Combinatorics, 9th Conference" }{TEXT -1 122 ", Universit\344t Wien, p.\240172-183, Conference Proceedings. Prelim inary version available as: Research Report n\260\2403195, INRIA." }} {PARA 15 "" 0 "" {TEXT 272 74 "Non-Commutative Elimination in Ore Alge bras Proves Multivariate Identities" }{TEXT -1 46 ", by Fr\351d\351ric Chyzak and Bruno Salvy, (1998), " }{TEXT 264 31 "Journal of Symbolic \+ Computation" }{TEXT -1 28 ", vol.\24026, n\260\2402, p.\240187-227." } }{PARA 15 "" 0 "" {TEXT 273 52 "Holonomic Systems and Automatic Proofs of Identities" }{TEXT -1 72 ", by Fr\351d\351ric Chyzak, (1994), Rese arch Report n\260\2402371, INRIA. 61\240pages.)" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 17 "Seminar Summaries" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "Holonomy is a recurrent topic discussed at the Algorithm Semin ar. Here are a few summaries of sessions dedicated to it." }}}{EXCHG {PARA 15 "" 0 "" {TEXT 274 52 "Introduction aux fonctions holonomes en une variable" }{TEXT -1 37 ", by Philippe Flajolet, (1992). In: " } {TEXT 296 29 "Algorithms Seminar, 1991-1992" }{TEXT -1 95 ", Ph. Flajo let and P. Zimmermann, (editors), (1992), Research Report n\260 1779, \+ INRIA. p. 41-45." }}{PARA 15 "" 0 "" {TEXT 275 41 "Fonctions holonome s \340 plusieurs variables" }{TEXT -1 33 ", by Kevin Compton, (1992). \+ In: " }{TEXT 297 29 "Algorithms Seminar, 1991-1992" }{TEXT -1 95 ", P h. Flajolet and P. Zimmermann, (editors), (1992), Research Report n \260 1779, INRIA. p. 47-49." }}{PARA 15 "" 0 "" {TEXT 276 29 "Holonom ic Symmetric Functions" }{TEXT -1 47 ", by Dominique Gouyou-Beauchamps , (1992). In: " }{TEXT 298 29 "Algorithms Seminar, 1991-1992" }{TEXT -1* 95 ", Ph. Flajolet and P. Zimmermann, (editors), (1992), Research R eport n\260 1779, INRIA. p. 51-55." }}{PARA 15 "" 0 "" {TEXT 277 52 " Holonomic Systems and Automatic Proofs of Identities" }{TEXT -1 35 ", \+ by Fr\351d\351ric Chyzak, (1995). In: " }{TEXT 288 29 "Algorithms Sem inar, 1994-1995" }{TEXT -1 72 ", B. Salvy, (editor),\n(1995), Research Report n\260 2669, INRIA. p. 39-42." }}{PARA 15 "" 0 "" {TEXT 278 37 "Creative Telescoping and Applications" }{TEXT -1 35 ", by Fr\351d \351ric Chyzak, (1996). In: " }{TEXT 292 29 "Algorithms Seminar, 1995 -1996" }{TEXT -1 72 ", B. Salvy, (editor), (1996), Research Report n \260 2992, INRIA. p. 39-42." }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "d;" "6 #%\"dG" }{TEXT 279 17 "-Finite Functions" }{TEXT -1 35 ", by Fr\351d \351ric Chyzak, (1996). In: " }{TEXT 293 29 "Algorithms Seminar, 1995 -1996" }{TEXT -1 72 ", B. Salvy, (editor), (1996), Research Report n \260 2992, INRIA. p. 43-46." }}{PARA 15 "" 0 "" {TEXT 280 53 "New Alg orithms for Definite Summation and +Integration" }{TEXT -1 35 ", by Fr \351d\351ric Chyzak, (1997). In: " }{TEXT 294 29 "Algorithms Seminar, 1996-1997" }{TEXT -1 72 ", B. Salvy, (editor), (1997), Research Repor t n\260 3267, INRIA. p. 27-30." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "Here are also seminar summa ries of sessions about related topics, either applications or algorith ms to be used internally in Mgfun." }}}{EXCHG {PARA 15 "" 0 "" {TEXT 281 99 "Un algorithme efficace pour le calcul des solutions rationnell es d'un syst\350me diff\351rentiel lin\351aire" }{TEXT -1 35 ", by Mou lay Barkatou, (1997). In: " }{TEXT 295 29 "Algorithms Seminar, 1996-1 997" }{TEXT -1 72 ", B. Salvy, (editor), (1997), Research Report n\260 3267, INRIA. p. 31-32." }}{PARA 15 "" 0 "" {TEXT 286 48 "Short and E asy Computer Proofs of Partition and " }{XPPEDIT 18 0 "q;" "6#%\"qG" } {TEXT 287 11 "-Identities" }{TEXT -1 31 ", by Peter Paule, (1995). In : " }{TEXT 289 29 "Algorithms Seminar, 1994-1995" }{TEXT -1 72 ,", B. S alvy, (editor),\n(1995), Research Report n\260 2669, INRIA. p. 43-46. " }}{PARA 15 "" 0 "" {TEXT 284 26 "Symbolic Computation with " } {XPPEDIT 18 0 "P;" "6#%\"PG" }{TEXT 285 17 "-finite Sequences" }{TEXT -1 35 ", by Marko Petkovsek, (1993). In: " }{TEXT 299 29 "Algorithms \+ Seminar, 1992-1993" }{TEXT -1 72 ", B. Salvy, (editor), (1992), Resear ch Report n\260 2130, INRIA. p. 51-54." }}{PARA 15 "" 0 "" {TEXT 283 49 "Polynomial Solutions of Linear Operator Equations" }{TEXT -1 35 ", by Marko Petkovsek, (1995). In: " }{TEXT 290 29 "Algorithms Seminar, 1994-1995" }{TEXT -1 72 ", B. Salvy, (editor),\n(1995), Research Repo rt n\260 2669, INRIA. p. 31-34." }}{PARA 15 "" 0 "" {TEXT 282 70 "Lin ear Recurrences, Linear Differential Equations and Fast Computation" } {TEXT -1 31 ", by Bruno Salvy, (1996). In: " }{TEXT 291 29 "Algorithm s Seminar, 1995-1996" }{TEXT -1 72 ", B. Salvy, (editor), (1996), Rese arch Report n\260 2992, INRIA. p. 31-38." }}}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 11 "regexpc-ount" }}{EXCHG {PARA 15 "" 0 "" {TEXT 302 40 "Re gular expressions into finite automata" }{TEXT -1 35 ", by A. Brueggem ann-Klein, (1993), " }{TEXT 303 28 "Theoretical Computer Science" } {TEXT -1 17 ", vol. 120, n\260 2." }}{PARA 15 "" 0 "" {TEXT 301 16 "Mo tif Statistics" }{TEXT -1 58 ", by P. Nicod\350me, B. Salvy, and P. Fl ajolet, (1999). In: " }{TEXT 300 50 "7th Annual European Symposium on Algorithms ESA'99" }{TEXT -1 9 ", Prague." }}}}}{MARK "0 0 0" 0 } {VIEWOPTS 1 1 0 3 2 1804 } 35 ", by Marko Petkovsek, (1995). In: " }{TEXT 290 29 "Algorithms Seminar, 1994-1995" }{TEXT -1 72 ", B. Salvy, (editor),\n(1995), Research Repo rt n\260 2669, INRIA. p. 31-34." }}{PARA 15 "" 0 "" {TEXT 282 70 "Lin ear Recurrences, Linear Differential Equations and Fast Computation" } {TEXT -1 31 ", by Bruno Salvy, (1996). In: " }{TEXT 291 29 "Algorithm s Seminar, 1995-1996" }{TEXT -1 72 ", B. Salvy, (editor), (1996), Rese arch Report n\260 2992, INRIA. p. 31-38." }}}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 11 "regexpc-rocŠ ½8Õ\Îjrocedur¶JÜ_ÎjrocesBkrocq:Ü_jÎj:y rocquencourtP rod Š W5˜@¼QÜ_Bkëlrodg ¼QÎjroduct.roebn½4¶JCNkwrofëlrofhÕ\rog  AÜ_rogramjroh¿ roiëlrojectkwrokwg¿ rolbfëlrom¿ ÎjBkòsrombergŠ rooÕ\roofÕ\Õ\review¿ †P¼Q-B¼QdÆÍ  ^jdedicat-B•rîxdeduc ÎjodefC"'3(‚QdefaIvdefauý\defaultG!.ç 'Ô+Ð0V1S:Â;¥=†Pý\§b†iŸpÁq sIvdefaultgà¿ "defbsgW5defend•rdefi¿  "àAÕ\ëlekdël functionalit#P C'‡/Õ\Êo!vkwð à autocomb autocombŠ autocomb¿ autocomb£ autocomb½algolibŒalgolibKalgolibµ% autocombÔ+algolib ,algolibW5 autocombÞ5 Groebner7algolib AalgolibàA autocomb†P autocomb‚Qalgolib¼Q autocombÕ\ autocomb ^algolibÌ^ Groebner_ Ore_algebraždalgolibÎj autocombBk autocombël autocombo autocombÊoalgolib•ralgolib salgolib!valgolib:y autocombprouhetprovCà¿ µ%3(‡/J@àA†PÎjBko•r!vkw:yprovabŠ proved Îj:yprovenprovidk'Š ½K'X9ˆ;ü;<¥=8@J@àA¤N¼Q‰\Õ\Ü_)`ÎjëlÊo!v:yuy prtodiffeqòsprudµ%prun3Ë-prunedg3Ë-/ÀDprunegfË-prwëlprz ¿ Õ\ps# Š ¿ "àA¼QÕ\ÎjopsepÕ\pseud*Opseudo#R6ˆ;ü;<*O ^Êo9qpsgëlpshëlpsi"psig"psilonW5Ü_òspsilong:ypsjÕ\psmjf"psnëlpsnuÕ\pssëlpssiÎjpsyÕ\pt#¿ "W5àAÕ\Îjëlòsevalb‡/½8†PBkyjbëlyjdÕ\yje¿ yjfÕ\yjkàyjllëlyjsqfdëlyjxëlyk¿ "Õ\ÎjëlykchrqnÕ\ykcuhÕ\ykf ¿ Õ\ykfduÕ\yklllo"yklqlfÕ\yknÕ\yknbk¿ ykqÕ\ykuÕ\ykudfÕ\ykudhÕ\ykudhfÕ\ykudzfÕ\ykuqëlykw ¿ ëlykxtq¿ yl'àŠ ¿ "W5*O!QÕ\ëlylcgnÕ\ylkW5ylnëlylncëlylor ¿ ½ylpn¿ ylqëlylsW5ylsrleoÕ\ylvf"ym¿ Õ\ëlymaëlen'àŠ èK-B¼Q-[Îj:y-[‰\ Õ\ý\å] ^Ì^à^_Ü_)``§b|cd*djÎj'BkëlÔmo}oÊo ypŸp9qÁq•ròs!v Ivkwîx:y"uy:y&decalphaunixmaplinputcourimathtimehyperlinkcommoutputpresbulletnormalheadutputconstrainpermutatprinciplincluionexclusphilippflajoletjanuarworksheetexplorariationsomeclassicalproblemcombinatorialanalysilikehederangemalsocallrencontrmenagdescriptthesproblemsborrowcomtetguestpartleavtheirhatshookcloakroomgrahatatgoodluckprobabilitnobodgetsbackhisownequivalestimatnumberwithoutfixedpointsaysingletoncyclenagnumbpossiblwayscanarrangengmarricouplaroundtablsuchmenwomenalternatbutnowomaseatnextherhusbandfactenumerationconstraintonsiderconcernsigmasigmagsuccessgapsdifferencbetweenconsecutelementigcorrespondsecondcaseindicvalumayaltakenmodulocyclicalsymbolicmethodenumeratcombinatoricwellcombstructpackagimplementbasedconceptdecomposabilitobjectdefinthusaccessiblframeworkhowevshownvarioutypetreatcombinagfunsystemeithimposallforcedbelongfinitsetomegaomegaÜ SbhelpfunctgfunlisttoserconvertlistintoseriseriestolistlisttolistseriestoserusagcallsequencgfparameternameoptionalsynopsidescriptthesprocedurtakeinputyieldaccordtheirviewcoefficientpowerreciprocalgeneratdefaultcreatwhoselementgivenargumsimilarreturnotheridentitthirdtreattypeoutputthoscorrespondfollowfunctionknownogfrevogflgdogflaplacegfrevegflgdegfordinarexponentialithcoefficioplogarithmicderivatusercandefinhisownmytypeofgfvariablmusttaylorparticularcannothavenegatexponentexamplwithnlnlisttoserxgksjcogseealsoalsosolvqglewÎjqgpÕ\qgrfÕ\qgsgpÕ\qgsqivig¿ qgwepiëlqgwv¿ qgyÎjqh ¿ Õ\ÎjëlqhbÕ\qhdÕ\qhf" qhffdufhglÕ\ qhffdufhglfÕ\qhh¿ qhnfqfÎjqhpëlqhpi¿ qhwŠ  qhypergeomdqi#àŠ ¿ "W5Õ\Îjël qiaëlqicëlqicr†Pqif¿ qifd¿ qifhgq¿ qih¿ qimëlqipëlqirsfuëlqislvëlqiwÕ\qiyëlqiz Š Õ\xnbÕ\xndëlxngfŠ xnhyÕ\xnkiciëlxnll"xnmÕ\xnqëlxnunëlxnvëlxoŠ ¿ Õ\xoeboàxofÕ\xoffŠ xoheÎjxoraÕ\ xordereqng¿ xosëlxp Š ¿ ïàAÕ\ëlxpand£ xpans ¿ £ xpcountË-/q8xpdëlxpfxpg †PÎjxplainòsxplicit A¼QÎjxppÎjxpqël { Ë- / 78 q8 ÀD ˜] •rt Š}J@˜@ AàA-BÀD”E.F»I¶J»MãMCN*O†P!Q‚Q¼QWVªYXZ‰\Õ\ý\˜]å] ^Ì^à^_Ü_§bd†ijÎjBkëlÔmo}oÊoŸp9qÁqòstIvkwîx:yŠ}"expect3iŠ ¿ ï"V1½4†P¼QÕ\Îjëlffofcq9ffofgëlffofgo9ffofgofJ@ffofpëlffoscuÕ\ffpÆŠ µ%ëlÂffpf¿ ÖÕ\ël:y ffpfftfhpµ%ffpqëlffq W5Õ\ël:yffqf ú+Õ\:yffqfbÕ\ffqvàffr¿ Õ\ël:yffrf:yffrfeqffrixëlffry¿ ffs¿ Õ\Îjëlfft ¿ Õ\ël:yfftf:yfftfd:yffu ÎjëlffufdvÕ\ffv "Õ\ëlffwëlffwfÕ\ffx ¿ "Õ\ëlffxf:yffy ¿ Õ\ffz:yfg×|æ.Š ¿ ç £ 95CÆì¯Í "µ% '3(Ô+ú+[,è/V1 ½4W5Þ57 S:ü;J@àA -BCN†P!Q‚Q¼QWV-[‰\Õ\+Ü_djÎjBkëlo9qt:yuytionalit#P C'‡/Õ\Êo!vkw}{TEé ãMÌdecalphaunixmaplinputcourimathtimehyperlinkoutputhelpnormalheadheadinlistitembulletencyclopediagfsearchsearchcombinatorialstructurdatabasegeneratfunctencyclopediacfsearchclosformusagcallsequencexprparameterwithsinglvariabldependsingleinfodescriptreturnstructursuchnumbobjectsizeallncyclopediafunctiongiveeachnamecombstructgrammarspecificatcombstructintegertermdefinfunctioncountdescribthlabellexponentialproducunlabelluniversordinarusedhisobtaingfsolvlinearrecurrencumberordernecessarholonomicgfunholexprtodiffeqdiffeqtorecdiffeqtorecthesnumberfirstasymptoticexpanscoefficienttaylorserineighbourhoodindextendinfinitotherwiscaseheydividsomereferencintegeiscontainnbavailablathttpwwwresearchattcomnjaseisonlinhtmlcanalsoaddresurlwebpagenohaveanswfailpossiblextractpartresultopexamplencyclopediaampermutatfrssetcyclhrlabellfrelementeulerstirlwordwordsgspec¤ •rÉdecalphaunixmathtimecommnormalheadbulletitemtitlauthorresearchpaperrelatalgoliblibrarsortrelevantpackagcombstructcalculurandomgeneratcombinatorialstructurflajoletzimmermancutsemtheoreticalcomputsciencvolgivebackgroundlabellcaseattributgrammarautomaticomplexitanalysimishnainriaresearchreportgrammarsusefulcombinatoricdelestfedouautomaticaveragalgorithmflajoletsalvtheoreticalserigagenerationpaulmapltechnicalnewslettearliverslambdaupsilonomegacookbookeportlambalambdagupsilongomegagluoshortpreviouincarnatassistantanalyzerapplialgebraalgebraicrrorcorrectcodemoraeditorlecturnoteobjectdutourdiscretmathematictheoreticaunlabellstructureeithnmurrasummarseminartalkbrunoresearcgdevexamplasymptoticexpanssigsambulletingfungfunmanipulatholonomicunctionvariablacmtransactmathematicalsoftwaroregroebnholonommgfunpublicatfollowdescribpectimplementfonctionencalculformelfrri9cchyzakthseuniversitairtudefendmaypagegrbnerbasesymbolicsummationintegratbuchbergwinklapplicatprocconferencyearcambridguniversitpreslondonsocietpreliminaravailablextenszeilbergfastgeneralfunctionformalpowerwienproceedpreliminarnoncommutateliminatalgebrasprovmultivariatidentitjournalcomputatsystemproofresearchrecurrtopicdiscussatseminarherefewsessiondedicatintroductauxunephilippphflajoletplusieurkevincomptonicsymmetricdominiqugouyoubeauchampsemcreattelescopdgfinitnewalgorithmdefinitreporalsosummariesabouteithalgorithmsusedinternalefficacpourledessolutionrationnelessystmediffrentiellinairemoulaybarkatouasypartitqgpeteralvywithpgsequencmarkopetkovsekpolynomiallinearoperatorequationreportearrecurrencdifferentialregexpcountregularexpressintoautomatabrueggemannkleinmotifstatisticnicodflajoletannualeuropeansymposiumesapraguementfonctionencalculformelfrri9û({VERSION 2 2 "HELP" "2.2"} {USTYLETAB } {SECT 0 {SECT 0 {PARA 0 "> " 0 "" {TEXT 26 10 "Function: " } {TEXT -1 66 "gfun[borel] - compute the Borel transform of a generating function" } } {PARA 0 "> " 0 "usage" {TEXT 26 17 "Calling Sequence:" } {TEXT -1 24 "\n borel(expr, a(n), t)" } } {PARA 0 "> " 0 "" {TEXT 26 11 "Parameters:" } {TEXT -1 4 "\n " } {TEXT 23 7 "expr - " } {TEXT -1 52 "a linear recurrence with polynomial coefficients\n " } {TEXT 23 7 "a,n - " } {TEXT -1 40 "the name and index of the recurrence\n " } {TEXT 23 7 "t - " } {TEXT -1 19 "(optional) 'diffeq'" } } } {SECT 0 {PARA 0 "> " 0 "synopsis" {TEXT 26 12 "Description:" } } {PARA 15 "> " 0 "" {TEXT -1 145 "If (a(n),n=0..infinity) is the sequence of numbers defined by the recurrence expr, the procedure computes the recurrence for the numbers a(n)/n!." } } {PARA 15 "> " 0 "" {TEXT -1 241 "If an optional argument 'diffeq' is given, expr is considered as a linear differential equation with polynomial coefficients for the funù !v­C{VERSION 3 0 "DEC ALPHA UNIX" "3.0" } {USTYLETAB {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 " " -1 256 "" 1 14 0 0 0 0 1 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 "" 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 }{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 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -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 <1 0 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 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 1 12 0 0 0 0 1 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 =0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 295 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 296 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Norm al" -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 "Bullet Item" 0 15 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 15 2 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 >0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 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 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 34 "Studies in Automatic Comb inatorics" }}{PARA 18 "" 0 "" {TEXT 288 37 "Combinatorics meets comput er algebra!" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 19 "" 0 "" {TEXT 256 18 "Algorithms Project" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 180 "Here is a series of notes that describe interactions between combinatorial analysis, discrete mathem atics, and computer algebra. 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 -1 279 ". The series appeared in two volumes, Vo lume I (1?996) and Volume II (1997), and are available from the web sit e of the Algorithms Project. The present worksheet lists case studies sorted by application domains, and also points to additional tutorial s and package introductions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Note that several of these worksheets have been produced with earlier releases of the " }{TEXT 293 7 "algolib" } {TEXT -1 96 " packages, and have not been updated. You may thus get d ifferent outputs when reproducing them." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Tutorials and In troductions" }}{EXCHG {PARA 15 "" 0 "" {HYPERLNK 17 "Introduction to t he 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 to 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 collection of " }{TEXT 258 10 "Combstruct" }{TEXT -1 184 " examples showing how to generate random trees, investigate th e distribution of height by simulation, enumerate functional graphs, a lcohols (!), necklaces, expression trees, and so on." }}{PARA 15 "" 0 "" {HYPERLNK 17 "The Combstruct Package, Generating Functions." 2 "combstruct/generating_functions" "" }{TEXT -1 31 " Starting with version 3.0 of " }{TEXT 259 10 " Combstruct" }{TEXT -1 120 ", it becomes possible to produce generating 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 intAegration 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 manipulaBtion 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 " Starting \+ with version 3.2 of " }{TEXT 263 10 "Combstruct" }{TEXT -1 87 ", it is possible to perform some simple complexity analyses of algorithms ope rating on " }{TEXT 264 10 "Combstruct" }{TEXT -1 29 " structures in th e spirit of " }{TEXT 265 3 "Luo" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {HYPERLNK 17 "CGenerating Marked Combstruct Grammars." 2 "combstruct/ma rk1" "" }{TEXT -1 322 " Version 3.2 contains new functions producing \+ grammars with marked objects. This makes it possible to analyse means and variance of parameters of various combinatorial objects. The bas ic mechanisms are described here together with examples like: cycles i n permutations, path length or leaves in binary trees, and so on." }} {PARA 15 "" 0 "" {HYPERLNK 17 "More Examples of Marking Combstruct Gra mmars." 2 "combstruct/mark2" "" }{TEXT -1 59 " This worksheet continu es to explore the possibilities of " }{TEXT 266 16 "combstruct[mark]" }{TEXT -1 213 ". How far is the common ancestor of two nodes in a ran dom binary tree? What is the average distance between two nodes? Thi s and a few other examples related to non-crossing configurations are \+ to be found there." }}{PARA 15 "" 0 "" {HYPERLNK 17 "Attribute Grammar s and Combinatorics." 2 "combstruct/attributes" "" }{TEXT -1 28 " Thi s worksheet introduces " }{TEXT 299 1D0 "Combstruct" }{TEXT -1 410 " ca pability for describing properties of structures, like pathlength of t rees, using attribute grammars. Since for some structures and algorit hms it is possible to define a property corresponding to the complexit y of the algorithm on the structure, these functions provide another m echanism of algorithm analysis. However, in this case there is 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 Combinatorial \+ Geometry." 2 "autocomb/noncrossing_1" "" }{TEXT -1 64 " There, starti ng with Euler's counting of triangulations of an " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 142 "-gon, many planar configurations can be count ed, listed and randomly generated, automatically. This sheet may serv e as an entry point to the " }{TEXT 272 10 "Combstruct" }{TEXT -1 5 " \+ and " }{TEXT 273 4 "Gfun" }{TEXT -1 10 " packages." }}{PARA E15 "" 0 " " {HYPERLNK 17 "Enumerating alcohols and other classes of chemical mol eculs, an example of Poly\341's theory." 2 "autocomb/alcohols" "" } {TEXT -1 85 " Classes of chemical compounds can be represented by com binatorial models using the " }{TEXT 274 10 "Combstruct" }{TEXT -1 9 " package." }}{PARA 15 "" 0 "" {HYPERLNK 17 "Balls and Urns, Etc." 2 "a utocomb/balls_and_urns" "" }{TEXT -1 211 " Balls and urns models are \+ basic in combinatorics, statistics, analysis of algorithms and statist ical physics. We demonstrate here how their properties can be explore d using most of the functionalities of the " }{TEXT 275 10 "Combstruct " }{TEXT -1 9 " package." }}{PARA 15 "" 0 "" {HYPERLNK 17 "Combinatori cs of Non-Crossing Configurations." 2 "autocomb/noncrossing_2" "" } {TEXT -1 294 " Take points on a circle and consider graphs based on t hese points such that no edges cross. A fairly complete theory of the se constrained random graphs can be developed. Planarity entails a ve ry strong combinatorFial decomposability that is especially well suited to a detailed treatment by " }{TEXT 267 10 "Combstruct" }{TEXT -1 1 " ." }}{PARA 15 "" 0 "" {HYPERLNK 17 "Constrained Permutations and the P rinciple of Inclusion-Exclusion." 2 "autocomb/permutations" "" }{TEXT -1 33 " This worksheet is based on the " }{TEXT 268 10 "Combstruct" } {TEXT -1 5 " and " }{TEXT 269 4 "Gfun" }{TEXT -1 318 " packages. It s hows how to enumerate many classes of permutations with constraints on ``succession gaps'' (differences between consecutive elements). This covers many celebrated combinatorial problems (like ``rencontre'' or \+ ``menage''). Generating functions, recurrences, and asymptotics are o btained automatically." }}{PARA 15 "" 0 "" {HYPERLNK 17 "Robustness in Random Interconnection Graphs." 2 "autocomb/random_graphs" "" }{TEXT -1 33 " This worksheet is based on the " }{TEXT 270 10 "Combstruct" } {TEXT -1 5 " and " }{TEXT 271 4 "Gfun" }{TEXT -1 165 " packages. It s hows how to characterize the trade-offs betwGeen the density of edges i n a graph, its connectivity by short paths, and its robustness to link failure." }}{PARA 15 "" 0 "" {HYPERLNK 17 "Monomer-Dimer Tilings." 2 "autocomb/polymer_tilings" "" }{TEXT -1 253 " The number of ways a sq uare lattice can be tiled with unit squares and dominoes is a combinat orial problem related to physical models of phase transition. This wo rksheet applies combstruct to finding bounds on the asymptotic behavio ur of this number." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "Analytic \+ Combinatorics" }}{EXCHG {PARA 15 "" 0 "" {HYPERLNK 17 "A Problem in St atistical Classification Theory." 2 "autocomb/hierarchy_trees" "" } {TEXT -1 136 " Classification theory makes use of classification tree s. This worksheet explores properties of random classification trees \+ using the " }{TEXT 276 10 "Combstruct" }{TEXT -1 13 " package and " } {TEXT 277 5 "Maple" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {HYPERLNK 17 "P ollard's Rho Algorithm." 2 "autocomb/Pollard_algo" "" }{TEXT -1 101 " H\+ An efficient and simple technique used to find factors of integers. \+ We show in this worksheet how " }{TEXT 278 10 "Combstruct" }{TEXT -1 5 " and " }{TEXT 279 4 "Gfun" }{TEXT -1 162 " can be used to analyze a realistic combinatorial model of the algorithm and thus derive a prob abilistic complexity analysis of this algorithm and variants of it." } }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 34 "Symbolic Summation and Integra tion" }}{EXCHG {PARA 15 "" 0 "" {HYPERLNK 17 "Gfun and the AGM." 2 "au tocomb/AGM" "" }{TEXT -1 144 " The arithmetic-geometric mean is relat ed to hypergeometric functions. This relation and a generalization of it are explored and proved using " }{TEXT 281 4 "gfun" }{TEXT -1 1 ". " }}{PARA 15 "" 0 "" {HYPERLNK 17 "Variations on the Sequence of Ap \351ry Numbers." 2 "autocomb/Apery_numbers" "" }{TEXT -1 131 " Findin g a second order recurrence satisfied by combinatorial numbers was a c rucial step in Ap\351ry's proof of the irrationality of " }{XPPEDIT 18 0 "Zeta(3)" "6#-%%ZetaG6I#\"\"$" }{TEXT -1 259 ". In this session, \+ we exemplify an algorithmic method for symbolic summation by rediscove ring the recurrence on these numbers and proving a combinatorial ident ity that they satisfy. We also derive an efficient calculation 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_integral" "" }{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 inte gral from a recent research paper. This nice integral involves the fo ur types of Bessel functions." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "Asymptotics" }}{EXCHG {PARA 15 "" 0 "" {HYPERLNK 17 "Patterns in W ords." 2 "autocomb/DNA" "" }{TEXT -1 50 " How likely is it that a spe cific pattern occurs " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 34 " tim es in a random word of length " }{XPPEDIT 18 0 "n" "6#%J\"nG" }{TEXT -1 98 "? How does the probability of this event depend on the specifi c pattern? This worksheet applies " }{TEXT 283 10 "Combstruct" } {TEXT -1 5 " and " }{TEXT 284 4 "Gfun" }{TEXT -1 77 " to this problem \+ which has a connection to questions from biology of the DNA." }}{PARA 15 "" 0 "" {HYPERLNK 17 "A Seating Arrangement Problem." 2 "autocomb/c hannel_allocation" "" }{TEXT -1 20 " What happens when " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 334 " persons arrive at a luncheonette an d are so unfriendly that no one wants to sit next to an already occupi ed seat? This worksheet explores properties of this problem when peop le arrive randomly. (This also serves as a simplified model of channe l occupation in mobile communication.) The complete treatment is enti rely based on the " }{TEXT 285 4 "Gfun" }{TEXT -1 13 " package and " } {TEXT 286 5 "Maple" }{TEXT -1 50 "'s capabilities in solving different ial equations." }}{PARA 15 "" 0 "" {HYPERLNK 17 "Staircase polygons, a simplified modelK for self-avoiding walks." 2 "autocomb/self_avoiding_ walks" "" }{TEXT -1 193 " There, we count pairs of non-crossing paths in integer lattices of dimensions 2, 3 and more. We also get numeric al asymptotics by a connection method. This sheet makes intensive use of the " }{TEXT 287 4 "Gfun" }{TEXT -1 9 " package." }}{PARA 15 "" 0 "" {HYPERLNK 17 "Asymptotics of the Stirling Numbers of the Second Kin d." 2 "autocomb/Stirling_numbers" "" }{TEXT -1 443 " The asymptotics \+ of the Bell numbers is a classical problem which is traditionally trea ted by the saddle-point method. The asymptotic scale required to perf orm the computation is non-trivial, and variants of this problem such \+ as the asymptotic behaviour of the average value or the variance of th e Stirling numbers involve an indefinite cancellation in this scale. \+ This worksheet exemplifies the use of a recent algorithm on this probl em. " }{TEXT 282 29 "[Based on experimental code.]" }}}}}{MARK "0 0 0 " 0 }{VIEWOPTS 1 1 0 3 2 1804 } modelKŠ xdecalphaunixmaplinputcourimathtimecommoutputnormalheadplotmonomdimertilingcazalsdecembfundamentalproblemlatticstatisticsiteregularcovernonoverlappmonomerdimerssquarpairneighborexamplsuchmxnmxngchesboardwithmgngdepictbelowrelatnumbcanarbitrarmayconstrainsomedensitpggeneralanyfixeddimensdgmodelintroduclongagoinvestigatpropertadsorbdiatomicmoleculcrystalsurfacrobdimensionalversoccutheormixturdiffersizegugwellcellclustliquidstatcoalpracticalmostthermodynamicthesphysicalsystemderivwaysgivenedconsiderablattentdevotcountquestprovabgoodpolynomialapproximatalgorithmexposkenalbutexactountresultstillunknownevencurvesgfcfbfcfafrfqfkfgfrfpfenfzfqfenfqfinflfqfcoffaoffcofbofaocolourgrgbgfscalinggconstrainedggoalworksheetshowamenablautomatcomputalgebratreatmentgoesspecificatcoveringsconstructtermcombstructgrammarasymptoticusingrationalgeneratfunctionnumerics˧bâdecalphaunixmaplinputcourimathtimehyperlinkoutputhelpnormalheadheadinbulletitemcombstructallstructlistallcombinatorialobjectgivensizeusagcallsequencspectypstructargsstructparameternonterminalspespecificatlabellingtypelabellunlabelldefaultnlabellnegatintegspecifyobjectstringallsizavailablonlypredefinstructurespossiblgeneratargumcorrespondinfodescriptcommandeturnelementclasrequirwillreturnexceptsometimsubsetcombinatsetspecifiusedmustworkwithgrammarcombstruciterstructcreatiteratorthesproducneelematraththemonceombstructexamplcombstrnallstructredblueredgbluegpermutatagbgcgprodpowersetprodgzgpowersetgepsilongseealsoalsonihilators,Ore_algebraˆ;applyopr,Ore_algebraS:balls_and_urns,autocomb¼Qchannel_allocation,autocomb†Pcommutation_rules,Ore_algebra9qkfnòskfpfcrfàAkft:ykfvbÕ\kgºàiŠ ë ¿ £ 9?Í "µ%3([,‡/7 S:àA-BÀD†P !Q‚Q¼QWVÕ\jÎjëloÊoòst!vkw:y kgeÎjkgfC!à¿ Í "3([,7àA-B†P‚QWVÎjëlkw:ykgfhÎjkggëlkghÎjkgjÕ\kgn Îj:ykgrkxàkgrpBkkh¿ W5Õ\ëlkhk¿ khllÕ\khovlÕ\khrfÕ\khruÕ\khtŠ khxÕ\ki ¿ ƼQÕ\ÎjëlkicÕ\kid"kieëlkiefg¿ kif .FÕ\kifkofÕ\kikcëlkilÕ\kin!vkind# Š £ "¼QÜ_jBkëlkiyneg¿ kj à¿ kjhÕ\o6decalphaunixmaplinputcourimathtimehyperlinkcommoutputnormalheadoutputitleauthorgfunagmbrunosalvjanuarletagbgpositrealnumberswitharithmeticgeometricmeanclassicaldefincommonlimitsequnceskgsqrtsqrtgnthatsequencconvergsamecaninferrnowngaussagmdiscovergausrelathypergeometrifunctionhypergeomgaussagmghypergeomgevalsubsoucoucfworksheetlargeinspirshowhowusedguesndprovresultwellgeneralizatdueborweinfunctionalequatfollowwestartintroducarithmetiobtainconsiderfollowiteratngintegngfsecondequalitmotivatbothsequencconvergdenotmggeomtriccorrespondcasefuncteasiseenhomogeneoulambdalambdagagfbgftogethobvioupropertimplxgxgfabovtranslatintofuneqfuneqgwhichplaycentralletoodifficultanalyticneighborhoodoriginuniqusoluteighborhoodquadratictheoremequivalstatnousefirststepcomputseriexpansguesspossiblclosformturndoessatisfopmethodundeterminatcoefficientsoldosolvuzfirtstÎjfis µ%Õ\ëlfisf:yfisffqf:yfit¼QÕ\ëlfitf:yfitfcÕ\fiu ¿ ëlfiud¿ fiuf:yfiv ¿ Õ\five ç "˜@ Aëlofivff:yfiw Õ\Îjfix P ¼QÎjëlfixed?Š ë 9Æ"Â;˜@¼Qý\Ü_ÎjBkël:yfixf:yfixing¼QfiyŠ Õ\ëlfiz ¿ Õ\ëlfizf ¿ " fizfkfjzf:yfjOàŠ ¿ ÖÍ "‡/W5ˆ;àA.F¼QÕ\%Îjël òsîx:yV1ƒ4½4Þ5R6778q8½8S:ˆ;Â;ü;<¥=ó=$@˜@ A-BÀD”E.F»I¶J»MãMCN*O!Q‚Q¼QWVªYXZý\˜]å] ^Ì^à^_Ü_§bdjÔm}oÊoŸp9qÁq•rt!vIvkwŠ}pwuëlpwvc¿ pwxwkëlpx à¿ Õ\ÎjëltIvkwîx:yŠ} distributi†Pdistributional†P disubstitut W5 ditionÍ diuÕ\divˆ;ü;Îjdiverg:ydivi‡/dividGài¿ £ Æk¿"‡/½8ˆ;<ãMÜ_BkÔmòsdivisˆ;<.F ^divisiå]division‡/ü;Êodivisor Í .Fdizëldj¿ "<†PÕ\ëldjdpkif"djfÕ\djo¿ djpÕ\djpfÕ\djrÕ\djs"djv"djy†Pdk ëlÊodkf Õ\Îjdkmfòsdkozàdkrhofrdëldksëldku¿ dkvÕ\dl¿ "Õ\ël dlcxÕ\dle"dljëldll"dlllÎjdlyëldm"W5Õ\ël:ydmcëldmfÕ\rger¿ rgf# £ q8<LD¼QÕ\_Š}rghfÕ\rghuW5rgs¼QrgtàArguëlrgumÎjrgwkgÎjrgxektqfÕ\rgxnÕ\rh¿ W5Õ\ÎjBkëlrhcfduÕ\rhcudfgëlrheW5rhf"rhh¿ rhhobmcÕ\rhjtÎjrhkvp¿ rhmdaëlrhnëlrho8à¿ ¼QÕ\jBk!vrhogà¿ ¼QÕ\ BkrhogfÕ\rhonÕ\rhonumÕ\rhonumsgÕ\rhoxëlrhpÎjrhqëlrhrj¿ rhs˜@rhwÕ\¿ P W5 AÎjoiciÕ\icient .£ icjcÕ\icnÎjicosagon”E descriptioIvdesignà¼Q!v:yndominocŠ ‚ ådecalphaunixmaplinputcourimathtimehyperlinkoutputourihelpnormalimesheadbulletitemregexpcountrandregexpgeneratrandomregularexpressstarformstarnormalformputformdrawworddrawwordlanguagdefinautomatonusagcallequencregconstructorprobaepsregexpgramsizeparameternamealphabetsetchoosunionproductsequencprobaepnumericoptionalparametinsertepsilonwithprobabilitexpresswithoutgrammarregularinfodescriptiongeneratbuiltoverparstreeincludleavfunctnormalformreturnnotarrsubexpressnullablcorrespondcontainemptalsoimplifnestcomplexitconstructnondeterministicfreeautomatarecognizberrsethalgorithmquadraticbrueggemankleinintofinitautomatatheoreticalcomputsciencexamplprodagatomgbgfrgsequencegprodgnotgfgrregexptogramgrguniongepsilonggffffbprodgagfsimplifstarnormalformsubsbgstarrnotgepsilongfstarnormalforunsequncegceagainseealsoalspecificatregexpcountcombstructdrabgeÕ\bgf‹|3i‚ ï{Í 'Ë-/W5R678ˆ;<˜@ ALD.F»MãM*O¼QÕ\ ^dÎjëloòst:ybgfgW5bgj¿ bgkq¿ bgl¿ bgmtqëlbgoëlbgyzÕ\bh¿ Õ\ÎjëlbhefëlbhkÕ\bhnÕ\bhqëlbhqnr¿ bhvÕ\bhx ¿ Õ\bi ¿ "-BÕ\ël:ybiasël bibliographobidden:ybieÕ\biggŠ Õ\òsbijeBkbiject "òsbilisticëlbilit Îj:ybim¿ binÍ  AÎjBk Ivbina à"binargNàŠ ç 9:k" 'ƒ4$@˜@ A†PªY‰\à^jÎjBkÔmòs!vîx:ydecalphaunixmaplinputcourimathtimehyperlinkoutputhelpnormalheadeadingbulletitemregexpcountautoerrortransformexactautomatonerrorautomatongramerrorusagcallsequencautogramregparametertablinternalspecificatgrammarnamnumballowerrorssubsetsubstinsdelinfodescriptgivendeterministicgrammarrecognizregularlanguagwordsetthfunctproducheateditdistancoriginalparametdescribconsideredyouwantconsidmarkovcasefirstndcomputautomarkovgrammarkovegexpcountcompletexternalautomatawarncommandprocesserrorsmatchcountwaitprovidprocedurregexptomatchesgramexptomatchesgramegexpstowaitgramregexpstowaitgramgeneralovermarkwillotgivecorrectresultexamplwithregexpcountatomauregexptoauprinttableginitgagtransgprunedgtruegalphabetgbgfliststatesgfinalgfepsilonaabaaberaualphabetgfiststatesgfdffeautotogramsubscombstructgfsolvunlabellzggrregexptogramgrernscombtructseealsoalsospecificatregexptoa <decalphaunixmaplinputcourimathtimehyperlinkcommoutputourihelpheadnormallistitembulletashfunctgroebnnormalfformpolynomialmoduloidealreducfractfreefullreductusagcallsequencwgwlwlgtgsgparameterpossibskewsetskewtermordertablshortdescriptptionalnamesynopsidescriptioncomputwithrespectbasidescribmorespecificalyieldcanonicalfogeneratgivenotherwordreturnremaindpseudodivisrespectcommousethesfunctionafterexplicitusinggbasicaseoptimizallowefficisyntaxthereforavoidoperandelementmusteithlinearoperatororealgebrauseddefinalgebrashiftdiffdiftermordcommutatermpolyerthprocesrgsuchsumwgfsumgwlgfrgfsamenoleadelemdividanyprocedurreturnwhilfourthoptionalargumprovidvaluputcorrespondhesepartpackagcanonlyperformcommandalwayaccesslongexamplngngbtdeggbgxgzgfygfnnormalfgbzgygwinadnntnwnreducproductdngfgleducdjagbgffefngfexpandgffhffkfofeffvxgfseLnowS=àŠ ¿ :¯Í "µ%àA†P¼QÕ\djÎjBkëlòs:ynownonoy Õ\np7¿ CV1W5ˆ;¶J†P1VXZÕ\ ^òs:ynpartit 'ý\npath‰\npbinBknpbingBknpermÎj npermutat 'ý\npf Õ\Îjnpmoëlnpmtf¿ npndxŠ npolÎj npoltodiffeq|c npoltorecè/npow˜@npr ànprintÕ\nprod ˜@npsànpt‰\npzg†Pnq‡/W5ˆ;Õ\å]ëlnqf ‡/ëlnqfbÕ\nqjh¿ nqomtf¿  nqpochhamm ,nqtiÕ\tly¼Qtmà¿ "àAÕ\ël tmarkovautoÀDtmbÕ\tmfëltmfdëltmm"tmmyhf"tmmyjf"tmproc ‡/½8tmprocg½8tmtf¿ tmxmÎjtn¿ Õ\ÎjëltnbckecŠ tncW5tnfëltngW5tnhpjàtniëltnmmmmmÕ\ tnmmmmmmmfÕ\ tnmmmmmtgfÕ\tnpyÕ\tnsëltnucfgxÎjtocomb!v tocompletË- todeterminizq8 todiffeqgÎjtofgÕ\encuëlency ŒãMencycŒÔm}oencyclŒencycloP ãM:yencyclopŒkãM encyclopedi ŒÎj encyclopedia7QàŠ P Œk ¿W5ãM ¼QÕ\Ôm }o :yendc¿à.Š ¿ £ Í µ%3(W57½8Â;àA†P¼QÕ\dÎj(Bk ël9qòs:yending iÎjendoÊo endomorphism *OÊoendpoint Õ\òsends 7:yeneeneral:yenerat+ iŒw"'¶JÕ\Ü_Îjëleneratingfunctionolog"energ¼Qenf Õ\ëlenfgfnÕ\enfianÕ\ength"'Ü_òsenlaÎjenlarg¿ ermlineardifferentialdifferencperatorturnwellgroebnimplementgeneralbuchbergdealidealtypemultivariatfeaturcalculatovercomplicatedgroundfieldsinglmanydersiiicommoninvariantvarietivfacilitchangorderingolynomialmgfuintendmanipulatlargclasspecialdgfinitholonomicfunctionespecialtheiruserorientinterfacholonomavoiddirectdesignclosspirititywiperformcomplexitanalysoperatingluomarkmarkcontainobjectmakemeanvariancparametervarioubasicmechanismtogethcyclpathlengthleavbinarmoregrammarcontinuesexplorpossibilitfarancestornoderandomaveragdistancthifewotherrelatnoncrossconfiguratfoundattributintroduccapabilitpropertpathlengthreessincalgorithmsdefincorrespondprovidanothechanismhowevaccesstatisticalinformatsuchhighmomentplanargeometrautocombnoncrostartingeulertriangulatgoncanautomaticalsheetserventralcoholclasschemicalmoleculpolytheorcompoundrepresentcombinatorialmodelballu[rnsutocombstatisticstatisticalphysicwedemonstratmostfunctionalitcombinatoricstakecirclconsidbasedhesenoedgecrosfaircompletseconstraindevelopplanaritentailverystrongdecomposabilitsuitdetailtreatmrinciplinclusexclushowsconstraintsuccessgapsconsecutelementcovercelebratproblemrencontrmenagrecurrencasymptoticbtainrobustnesinterconnectcharacteriztradoffsdensitconnectivitshortlinkfailurmonomdimertilingpolymnumbwayssquarelattictiledunitsquardominoorialphysicalphastransitworksheetapplfindboundbehaviouranalyticstatisticalclassificathierarchollardrhopollardalgoefficitechniquusedfactorintegeranalyzrealisticderivprobabilisticvariantintegraagmautocombarithmeticgeometrichypergeometricgeneralizatprovvariatapnumberaperfindinsecondordersatisfirucialstepproofirrationalitzetazetagsessexemplifalgorithmicmethodrediscovringidentsatisffirsthundreddigitintegralproductfourbesselillustratco\mputatformintegralrecentresearchpaperniceinvolvfopatternordsdnaspecificoccurkgtimworddoesprobabiliteventdependspecificonnectquestionbiologseatarrangemhannelallocathappenpersonarrivatluncheonettunfriendwantnextalreadoccupipeoplesimplifichannoccupatmobilcommunicatentirelycapabilitdifferialstaircaspolygonselfwalkpairintegdimensnumericintensstirlkinbellclassicaltraditionaltreatedsaddlscalrequirperformtrivialbehaviourvaluindefinitcancellatproblemexperimentalcodeworksheetapplfindboundbehaviouranalyticstatisticalclassificathierarchollardrhopollardalgoefficitechniquusedfactorintegeranalyzrealisticderivprobabilisticvariantintegraagmautocombarithmeticgeometrichypergeometricgeneralizatprovvariatapnumberaperfindinsecondordersatisfirucialstepproofirrationalitzetazetagsessexemplifalgorithmicmethodrediscovringidentsatisffirsthundreddigitintegralproductfourbesselillustratco\oTX{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 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 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 "Maple Out put" 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 "T itle" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 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 } {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 " }{XPPED`IT 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*\"\"#!\"\"*$*&,&&F&6#F)F*&F/6#F)F4F*\"\"#F4\"\"#" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "This common limit is k nown by Maple as " }{HYPERLNK 17 "GaussAGM" a2 "GaussAGM" "" }{XPPEDIT 18 0 "``(a,b)" "-%!G6$%\"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 obtaibned 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 "cN=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-*&d,&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$\"\"\"),& \"\"\"F,F)!\"\"*&\"\"\"F,F'F6F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "the equation above translates into the following " }{TEXT 259 19 " functional equation" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "A[N](x)" "-&% \"AG6#%\"NG6#%\"xG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "funeq:=(1+(N-1)*x)*A[N](x^N)=A[N](1-((1-x)/(1+(N-1)*x)) ^N):" ">%&funeqG/*&,&\"\"\"\"\"\"*&,&%\"NGF(\"\"\"!\"\"F(%\"xGF(F(F(-& %\"AG6#F+6#)F.F+F(-&F16#F+6#,&\"\"\"F()*&,&\"\"\"F(F.F-F(,&\"\"\"F(*&, &F+F(\"\"\"F-F(F.F(F(F-F+F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "wh ich plays a central r\364le in this worksheet. It is not too difficult to show that " }{XPPEDIT 18 0 "A[N](x)" "-&%\"AG6#%\"NG6#%e\"xG" } {TEXT -1 139 " is analytic in the neighborhood of the origin and that \+ the functional equation above has a unique analytic solution in this n eighborhood. " }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "The quadratic \+ case" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "This is the case " } {XPPEDIT 18 0 "N=2" "/%\"NG\"\"#" }{TEXT -1 51 " and Gauss's theorem i s equivalent to stating that " }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[2](x)=hypergeom([1/2,1/2],[1],x)" "/-&%\"AG6#\" \"#6#%\"xG-%*hypergeomG6%7$*&\"\"\"\"\"\"\"\"#!\"\"*&\"\"\"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 analyticf 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>(og*$F%\"#6#\"+Tg`vx \"-Wp!z([F*$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 recurrenche 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 ",i 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#*$F(\"\"#F'F'-F*6#,&F'F '*&,&F'F'F(!\"\"F-F&!\"#F3F3\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "Our approach consists in using closure properties of solutions of linear differential equations that are implemented in " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 284 " to compute a linear differential equation satisfied by the left-hand side of this equation. The proof \+ then reduces to showing that 0 is the only solution of this differenti al equation that is compatible with the initial conditions, which are \+ 0 up to a large order by construction of " }{XPPEDIT 18 0 "y" "I\"yG6 \"" }{TEXT -1 144 ".\nIt turns out that this proof can be performed di rectly from the differential equation, and would apply even if no clos ed-form had been found. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Given a linear diffjerential equation satisfied by a series " }{XPPEDIT 18 0 "y(x)" "-%\"yG6#%\"xG" }{TEXT -1 15 ", the function " }{HYPERLNK 17 "gfun[algebraicsubs]" 2 "gfun[algebraicsubs]" "" }{TEXT -1 55 " comput es a linear differential equation satisfied by " }{XPPEDIT 18 0 "y(f( x))" "-%\"yG6#-%\"fG6#%\"xG" }{TEXT -1 51 " for any algebraic function , given by a polynomial " }{XPPEDIT 18 0 "P" "I\"PG6\"" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "P(x,f(x))=0" "/-%\"PG6$%\"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!\"\"\"\"#*$,&\"\"\"F'F+F'\"\"#F,F," }{TEXT -1 43 " is easily c omputed from that satisfied by " }{XPPEDIT 18 0 "y(x)" "-%\"yG6#%\"xG " }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "deq:=op (select(has,deq,x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "deq 1:=gfun[algebraicsubs](deq,numer(y-(1-(1-x)^2/(1+x)^2)),y(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%deq1G,(*&,&!\"\"k\"\"\"%\"xGF)F)-%\" yG6#F*F)F)*&,**$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$F4F)F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 165 "From there, we deduce a differential equation satisfied \+ by the left-hanld side of the functional equation when applied to the h ypergeometric function we have guessed:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "gfun[`diffeq+diffeq`](deq1,deq2,y(x));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,(*&,&\"\"\"F&%\"xG!\"\"F&-%\"yG6#F'F&F&*&,*F'F& *$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 1m4 "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 0n 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 difoferential 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\"\"\"#\"\"#\"\"$F*F*-F&6#,&F)F*#F*F-F*F*-F&6#,&F)F*F*F*!\"#%# PiG!\"\"F-#F*F,F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "y(x)=s um(\"*x^n,n=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6# %\"xG-%*hypergeomG6%7$#\"\"\"\"\"$#\"\"#F.7#F-F'" }p}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 31 "Proving the result of the guess" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "The same routine applies:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(A=y,funeq3)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,&\"\"\"F'%\"xG\"\"#F'-%\"yG6#*$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$F1F(F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4q7 "deq2:=gfun[`diffeq*diffeq`](\",y(x)+1+2*x,y(x)); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%deq2G,(*&,(\"\"#\"\"\"%\"xG!\"% *$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*/-Fr-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, Petsr 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, Pets hypothesi Õ\òshyr¼Qhyt ¿ ÎjhytràAhytrxhàAhyv¿ hyvno¿ hywëlhywjÕ\hz#àŠ ¿ W5†P¼QÕ\ëlhzaxëlhzenffgÕ\hzf W5ëlhzfbëlhzh†PhzhlpÎjhzni¿ hznoëlhzo"hzpeàhzswëlhztÕ\hzyëlia ¿ P k"Õ\Îjëliablîxiafëliagonal ë òsiai¿ ial;¿ ú+ ,V1àA»I‰\ ^dBkÊo9q!v:yength"'Ü_òsenlaÎjenlarg¿ seenŠ Õ\Îjo:yseesÕ\sef"sefuàAsegm¼QÎj:ysegnÕ\sei "Õ\select+¿ µ%‡/½8J@Õ\ ^Bkëloself ¿ !vÕ\ÌRdecalphaunixmaplinputcourimathtimecommoutputnormaltextoutputheadwarnerrorplotcombinatoricnoncrossconfiguratcazalaugustgeneralittakengpointequalspacunitcircldrawchordsbetweentheswithconstraintnochordcrosneanothresultcallstudsuchentitiesoriginatworkeulersegncounttriangulatgonsincseveraltypeconfigurationshavedefinexamplpresencabsenccyclnumbconnectcomponentgivenclasstreeforestgraphgeneraladditcombinatorialinterestperseconfigurationalsoimportantalgorithmicproblemariscomputgraphiccomputationalgeometrtheyprovidsimplmodelforealworldsituathistoricalcarrioutatturnallfitalgebraicanalyticthusamenableunifitreatmmoreprecisecantermsgrammargeneratfunctionautomaticalobtainusedasymptoticalanalysetcgoalworkshetpreschainspecificatusingcombstructasymptoticsomefeaturgfuntogethrandomsingularitdeterminbehaviourreadreferredflanodetailworksheetorganizfollowrestsectwïfiaëlfialÕ\fiapÕ\fib ŒÖ è/fiblëlfibmÕ\fibnÕ\ fibonacciŠ ŒKfibqÕ\fibrÕ\fibsÕ\fibuÕ\ficëlficatà$@˜]jficb†Pfice†PfichccjhÕ\fici £ ëlficient ü;ficl Õ\ëlficmÕ\ficoÕ\fidlëlfidlfÕ\fidnÕ\fidoÕ\fidqÕ\fidqfafqÕ\fied¿ field.‡/‚Q ^!vfiemÕ\fienÕ\fieqfÕ\ëltak-Btakayama#µ%3( 7J@àA-B‚Qdtakeo5àŠ ¿ £ Ö'‡/W5½8$@J@-B.F¶J†P¼Q S‰\Õ\ý\ÎjBkël!vîx:ytaken'àŠ èK-B¼Q-[Îj:yvq8=decalphaunixmaplinputcourimathtimehyperlinkoutputourihelpnormalimesheadbulletitemregexpcountautodeterminizdeterminizautomatonusagcallsequencautodeterminizautoparametertablinternalspecificatinfodescriptfunctreturndeterministicfinitequivalnonbothautomataspecifiinternalformregexpcountcompletepsilontransittheysuppresedbeforprocessdeterminizatnotestandardlogalgorithmminimizatirequirautominimizexamplwithregexpcountecognizlanguagaggfalphabetliststatinittranfinaldfadfagtableginitgtransgalphabetgffinalgfliststatesgcheckedgtruegtranslateintoexternalgrammargdfaautotogramgdfagatomguniongepsilongprodgrgfomputgeneratexpandfirsttermssubscombstructgfsolveunlabellserizgogalsoregexpcountutocompletautocompletregexpcounomb†Pcommutation_rules,Ore_algebra9qfhfoÕ\fhfq ëltfhfuf½4fhfuflëlfhg:yfhgnÕ\fhgoÕ\fhgpÕ\fhgpfaipÕ\fhgq¿ fhgsÕ\fhhlëlfhhnÕ\fhhqÕ\fhhrfbgrÕ\fhhs"fhhtÕ\fhi†Pfhil Õ\ëlfhimÕ\fhirÕ\fhitfÕ\fhjlfÕ\fhjnÕ\fhjqÕ\fhjsÕ\fhjuÕ\fhkÕ\fhkyÕ\fhn ¿ J@Õ\ël:yfhnf# ½4½8J@Õ\ ^Bkt:yfhnfin‰\ fhnfkfgnfxffn½fhnfoëlfhnfrypfhnfzfhalÕ\7Â;.FÕ\Bkël :ysfatff:ysfbgo" sfbwajhfgëlsfcfeqf:ysfdgp"sfdimÎjsfduÕ\sff Õ\:yBk(decalphaunixmaplinputcourimathtimehyperlinkcommoutputnormalheadutputpollardrhoalgorithmrunosalvversjanuarollardrhogmethodefficitechniquusedfindfactorintegerbothveryverysimplweshowworksheethowcombstructgfuncananalyzrealisticcombinatorialmodelthusderivprobabilisticcomplexitanalysivariantwithcombinatorialngintegsoughtbasicfollowpickuparbitraragmodsetfgxgmodgagfselectatrandomnumbygiteratiguntilgcdgcdgdirecttranslatintoroughprocedurreturnnumberperformprocnlocalrndranddoigcdfiodnendherefewexamplnextprimenextprimroutinsmallestprimargumcombinatorialrelitructurfunctionalgraphfunctpgverticedgelineachverteximagsincpointfinitdifficultstructurunionconnectcomponentsthescomponentconstitutcyclconvergtreepolynomialdegreallexcepthavewhiltheyoutassumlargthereforspecialcasediscardfirstapproximatexpressgrammarspecificatfungraphcomponprodbintrecardheexecut nappodprhltsutdsuzhnpulonztgvxwpgqxguessdifferentialsatisfideqseriestodiffeqogfdeqgygdiffgdgdsolvunablediffeqtorecdiffeqtorecdeducequatiorecurrenctaylorsolutionugfromorderlinearfoundrsolvgammagpighencsuminfinitypergeomgproofconsistobviouslourapproachusingclosurequationimplementlefthandsidereduconlydifferentialcompatiblinitialcondituplargconstructnitperformdirectapplevenedhadgivenalgebraicsubesfganyalgebraicpolynomialpgsuchthusomputselectnumersimilarproductdiffeqappliypergeometrichavelutioncoefficicgfffhfezeroconcludcubichypergeometricexpressalsoexistagainstepsleaderieheuneqtermthuqeowygeuxnhzwgvqblpiplmxmlucksituatfindbeforeumroutinffgdiffeqfunyfgfdffwfcfenfgnfaoffiveconclusionthesverygooexamplexperimentconjecturgeneralutionproblemcompletedifferproceshowevappareasetreatherehidepreliminarworkledunderseemhighvalumightbibliographgoeme{tricrevisitjonathanborweinpetrlisonekjohnmacdonaldmapletechppsfideqseriestodiffeqogfdeqgygdiffgdgdsolvunablediffeqtorecdiffeqtorecdeducequatiorecurrenctaylorsolutionugfromorderlinearfoundrsolvgammagpighencsuminfinitypergeomgproofconsistobviouslourapproachusingclosurequationimplementlefthandsidereduconlydifferentialcompatiblinitialcondituplargconstructnitperformdirectapplevenedhadgivenalgebraicsubesfganyalgebraicpolynomialpgsuchthusomputselectnumersimilarproductdiffeqappliypergeometrichavelutioncoefficicgfffhfezeroconcludcubichypergeometricexpressalsoexistagainstepsleaderieheuneqtermthuqeowygeuxnhzwgvqblpiplmxmlucksituatfindbeforeumroutinffgdiffeqfunyfgfdffwfcfenfgnfaoffiveconclusionthesverygooexamplexperimentconjecturgeneralutionproblemcompletedifferproceshowevappareasetreatherehidepreliminarworkledunderseemhighvalumightbibliographgoeme{Š {È{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 "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 "" 1 20 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 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 "" 20 262 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 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 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 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 "Headi}ng 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 {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]" "&%\"TƒG6#%\"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 corre„spond 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 constructi…on 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 2†75 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 Comˆbstruct[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#\"\"&\"$L(\"\"'\"%cB\"\"(\"%tv\"\")\"&UV# \"\"*\"&V#y\"#5-%\"OG6#F%\"#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 147 "This sequence does not appear in [Sloa95]. It can be checked that the se values match those computed directly from the grammer by Combstruct [count]:" }}}{EXCHGŠ {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(count([MM,G r2], size=i), i=0..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-\"\"\"\"\"# \"\"(\"#A\"#r\"$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 "Œ" {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" "$\"#J!\"#" }{TEXT -1 45 " so that the main contr ibution is encoded by:" }}}{EXCHŽG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "e l1:=subs(_alpha=root1,el);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$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 m‘ore 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!\"\"$\"+d6iB*)!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 142 "The number of occurrences and the proportions of dimer•s and mo nomers can be computed in the same way by erasing the irrelevant indet erminates:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 270 "pieceProport ion:=proc(MGF, keptPiece)\n local forSubs, stij, sstij, nbp;\n\n forSu bs:=\{h=1,v=1,c=1\} minus \{keptPiece=1\}; \n stij:=subs([op(forSubs)] , MGF);\n sstij:=subs(keptPiece=1, diff(stij,keptPiece));\n nbp:=evalf (ratasympt(sstij,z,n,1));\n asympt(nbp/nBDPiecesN,n,2)\nend:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "And we end up with:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "nbh:=pieceProportion(GF2,h);\nnbv:= pieceProportion(GF2,v);\nnbc:=pieceProportion(GF2,c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$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#*$%\"nG!\"\"F)" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 25 "Plotting routines archive" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "The figures above were pl–otted with the following functions:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 203 "dominoH:=proc(x,y) [[x,y], \+ [x+2,y], [x+2,y+1], [x,y+1], [x,y]] end:\ndominoV:=proc(x,y) [[x,y], [ x+1,y], [x+1,y+2], [x,y+2], [x,y]] end:\ndominoC:=proc(x,y) [[x,y], [x +1,y], [x+1,y+1], [x,y+1], [x,y]] end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 238 "plot([dominoV(0,0), dominoC(0,2),dominoC(0,3),\n \+ dominoC(1,0),dominoV(1,1),dominoH(1,3),\n dominoV(2,0),dominoC( 2,2), \n dominoC(3,0),dominoC(3,1),dominoV(3,2),\n dominoH(0 ,4),dominoH(2,4)],scaling=constrained,color=blue);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "plot([dominoV(0,0), dominoC(1,0),dominoV(1,1),dominoC(0,2),domino H(0,3),dominoV(0,4),dominoV(1,4)], scaling=constrained,color=blue);" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 39 "Automatic counting in a slice of width " }{XPPEDIT 18 0 "m" "I\"mG 6\"" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 34 "Computing th—e generating f unctions" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "We now show how to aut omate the previous computations for any integer " }{XPPEDIT 18 0 "m" " I\"mG6\"" }{TEXT -1 44 ". The first task consists in generating the " }{XPPEDIT 18 0 "2^m-1" ",&)\"\"#%\"mG\"\"\"\"\"\"!\"\"" }{TEXT -1 30 " words on the binary alphabet " }{XPPEDIT 18 0 "\{M,P\}" "<$%\"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%$MMMG%$PPM˜G%$MPPG%$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 lett™er 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(patteršn, 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.% $PPMGF1/F/-Fœ)6'-F,6%F.%$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 recComeŸsFrom(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)*$F&\"\"#F/F)" }{TEX¨T -1 7 ", then " }{XPPEDIT 18 0 "2*u[2 *n]-u[n]" ",&*&\"\"#\"\"\"&%\"uG6#*&\"\"#F%%\"nGF%F%F%&F'6#F+!\"\"" } {TEXT -1 4 " is " }{XPPEDIT 18 0 "l+O(1/n^2)" ",&%\"lG\"\"\"-%\"OG6#*& \"\"\"F$*$%\"nG\"\"#!\"\"F$" }{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+*$F,\"\")!\"\"F4!# BF9\"#HF6\"#\"*F " 0 "" {MPLTEXT 1 0 347 "gf[5]:=-(z^18+2*z^17-45*z^16-68*z^15+654*z^14+8 70*z^13-3820*z^12-4700*z^11+9255*z^10+9448*z^9-11175*z^8-7532*z^7+6956 *z^6+1994*z^5-1794*z^4-ª88*z^3+113*z^2+6*z-1)/(z^20+2*z^19-65*z^18-140* z^17+1281*z^16+2538*z^15-10366*z^14-17604*z^13+38553*z^12+50158*z^11-7 3623*z^10-60482*z^9+74665*z^8+26564*z^7-35106*z^6-898*z^5+4757*z^4+16* z^3-229*z^2-14*z+1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%#gfG6#\"\"& ,$*&,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!$H#F,!#9F.F.FgnFgn" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 882 "gf[6]:=-(-1+311*z^2-3891*z^3-12057 *z^4-315889*z^6-2997721*z^7+218447*z^5+13467571*z^9+8754480*z^8+23*z-4 58919487*z^18-303976032*z^17+612805499*z^16+207743591*z^15-496137395*z ^14-56233657*z^13+240612231*z^12-14684235*z^11-66016499*z^10+206819317 *z^20+249194245*z^19-109*z^32-36273*z^29+861*z^31+744«3809*z^24+3722360 1*z^23-123372421*z^21-54160427*z^22-6708699*z^25+z^34-29377*z^28+68651 7*z^27-338040*z^26+3521*z^30-7*z^33)/(1-576*z^2+6080*z^3+42422*z^4-443 404*z^6+12931566*z^7-453004*z^5-83558644*z^9-25517604*z^8-36*z+4169343 006*z^18+2978277152*z^17-4669345206*z^16-1630080704*z^15+3235975264*z^ 14+274712602*z^13-1335612340*z^12+154307596*z^11+295510396*z^10-231032 7672*z^20-2919950172*z^19+5736*z^32+1503868*z^29-62874*z^31-149620588* z^24-626694028*z^23+1717916424*z^21+777289050*z^22+141424642*z^25-8*z^ 35-138*z^34+z^36-94620*z^28-19237868*z^27+13835164*z^26-81796*z^30+122 4*z^33);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%#gfG6#\"\"',$*&,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 dim®ensional 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®µ%ä decalphaunixmaplinputcourimathtimehyperlinkcommoutputnormalheadoutputitleauthorintegralproductfourbesselfunctionfrricchyzakversjanuarglassmontaldisomeinvolvanalapplmontaldicomputclosformntegralsuggesttheirreatmextendfollowexamplintnfinitlnpiintgxgjgagfigygkginfinityglngfopigfofofointerestbecauscontaineachfoutypenumerouintegralsmoreinstancprudnikovbrychkovyumarichevserivolumspecialgordonbreachsecsesswedealwithabovderiveusingourmgfunpackagintimatinteractwitgfundiagsysgpolsumsyslaplacegalgebraicsubsgalgeqtodiffeqgalgeqtoseriesgalgfuntoalgeqgborelgcauchyproductgdiffeqdiffeqgdiffeqtohomdiffeqgdiffeqtorecgguesseqngguessgfghadamardproductgholexprtoinvborelglisttoalgeqglisttodiffeqglisttohypergeomglisttolistglisttoratpolyglisttorecglisttoseriesglisttoserlaplacegegfglgdegfglgdogfgisttoserogfgrevegfgrevogfgmaxdegcoeffgmaxdegeqngmaxordereqngmindegcoeffgmindegeqngm_ogj¿ ognizË-ogpudÕ\ogqëloh¿ W5àAÕ\Bkëlîxohadëlohaf¿ ohbÕ\ohcëlohe¿ ohfim kBkohgW5ohgfxuÕ\ohh †Pëlohhag¿ ohlÕ\ohliëlohllëlohlux¿ ohojëlohqÕ\ohu"ohumxfëlohx¿ oi ¿ ÖÕ\ÎjBkël:yoiaëloidwÎjoif Õ\ëloig"oikÕ\oilll"ointg†PoiogÕ\oirfëloirfdëloreg µ%ëlorestsgÕ\orevŠ orf"org ½8organizÕ\organoW5orgg½8orhood£ orhzÎjori Õ\Bkorial ABk!vorialg àÎjoricorient ¿ P 7àAÎj!vorientat¿ orig:yorigi †PÎjorigin+à¿ 3(71VÕ\ÎjBkooriginalCà¿ £ :ï"6½8ÀD.F†P¼QÎjBkòs:yoriginatÕ\orithm ˆ;•rorkorky†Porlajniëlorlv¿ orm Îj!v"gppÕ\gppnÕ\gpro ¿ ëlgprobëlgprobgëlgpv"gq¿ W5Õ\ëlgqbfw¿ gqi¿ gqlaw¼QgqqÕ\gqqmgëlgqqyëlgqqyyëlgqt ëluggiÕ\ugh†PughtÕ\uglëluglcÕ\ugnÕ\ugr.ugsàugzëluhàŠ ¿ Õ\ëluhduëluheef"uhegf"uhgfÕ\uhqjëluhrÕ\ui W5Õ\uicëluif¿ uild:yuiltuipÕ\uiredÜ_uisÕ\uishabl¼QuiuÕ\uiv¼Quivfëluix"gyf ¿ ëlgyfduÕ\gyfzybŠ gyloscÕ\gynzëlgyoÎjgypfëlgypoŠ gyqÕ\gyrqsmÕ\gysÕ\gyskëlgzŠ ¿ "Õ\ÎjëlgzdëlgzgzëlgzmëlgzmtjŠ gzoëlgzyfëlha'Š ¿ "W5.FÕ\Îj ëlòshaaoëlhabethack¼QhactëlhadBkoòshadama:yhadamard*dhadamardproduct K`+*dhadamardproductg¿ µ%àA†PÎjhafaÕ\picturëlpieŠ piec Š Îj pieceproportŠ pierrP pig/+à¿ £ "†P)`Bk oyp sòs pigf3à£ Í "µ%†PÕ\ždBkÊoòspigfoµ%pijÎjpiling¼QpipëlpiqelÎjpisa:ypitëlpitebÕ\piwëlpixeÕ\pizyoexÕ\pj ¿ "Õ\ëlpjex Š àApjfduÕ\pjfhÕ\pjhëlpji Õ\ëlpjlrwmtf¿ pjqmëlpjr¿ pjsÕ\pjyëlpjz Õ\pk "Õ\Îjëlpketuqhzëlpkf ¿ Õ\pkhnÕ\pkll"pkmtf¿ pkmwxëlëlgixÕ\giyÎjgjŠ ¿ "¼QÕ\ÎjëlÕ\sff Õ\:yuafÕ\uage ël:yëlTH 2* recursivelred¿ 9'ó= A§bîxredefin ç ˆ; redescrib"redg9§bîxredgf9' Aîxrediscov ¼Q!v rediscover†Preduc?Aèw"µ%‡/ˆ;ü;<ó=¼Q ^ BkoÊokwŠ}reduceg‡/ reducescalˆ;reduct<†P¼Qree ¿ ç :"'òsîxreegfjreen¿ reengf9rees!vref ‡/Õ\V1½4W57½8Â;$@J@ AàA-B†P‚QWVªY‰\Õ\ ^à^Ü_d*djÎjBko•ròs!v:yŠ} equatiosn†PQ¼Q S ªY‰\ Õ\ à^Ü_)`§b†ij Îj Bk ëlÔm}oÊoÁq•ròst!vkwîx :yuyexpansiongÕ\expcount3w"˜]Š}expeBkexpec"expect3iŠ ¿ ï"V1½4†P¼QÕ\Îjël"W5†Pdevic:yKxhelpgeneratfunctpackagusagcallsequencgfunargssynopsidescriptdesignmanipulatdiscoverfunctionprovidtoolcomputwithdefinequationinstancgivenlineardifferentialpolynomialcoefficientprocedurequatsatisfitheirproductdealrecurrencalgfuntoalgeqalgebraicsubalgeqtodiffeqalgeqtoserborelcauchyproductdiffeqdiffeqtohomdiffeqdiffeqtorechadamardproductholexprtodiffeqinvborellaplacpoltodiffeqpoltorecratpolytocoeffrecrectodiffeqrectohomrecrectoprocfirsttermalsocontainwillconjectursomecaseanswexplicitmostthoughsuchexpressexisteithalgebraicnumberseriguesseqnguessgflisttoalgeqlisttodiffeqlisttohypergeomlisttolistlisttoratpolisttoreclisttoserseriestoalgeqseriestodiffeqseriestohypergeomseriestolistseriestoratposeriestorecseriestoserparticulartakenabovlistexamplusingfollowfibonacciusecommandinformataboutcomputatdonecanobtainsettinfolevelanythbetweenmoreearlversfoundmaplholonomicvariablsalvzimmermanacmtranÜ agfe $@‰\agfeqnç 9Æ$@ A‰\Ü_agfeqnsg:' AÜ_agfmomÜ_ agfmoment‰\ agfmomentsolvç $@ A”E‰\Ü_agfmomentsolveg:' Aagfs Aagfserç 9 A‰\ agfseriesg:'Ü_aggregatàaglÎjagm o!vagnb¿ agoŠ agpoÕ\agqxlkmëlagrammar$@agreŠ ¿ àAagtyëlagu¿ agvëlah àÕ\ahatqmtf¿ ahew¿ ahgÕ\ahkÕ\ahmlnhqël interestingÍ interfacP '7!v interferenc†P intermediat:èÎjinterninterna ½W5internalcC3àç 9:"'Ô+Ë-/W578q8$@ÀD‰\˜]Ü_jBk•rŠ}interplaÎj interpret¿ ¼QBk9q:y interpretat‰\ intersect 3!QBkòskw intersectg!Qinterupt$@interva -BÎjinterval ¼QÎjîxinterven $@Îjintg/¿ µ%3(‡/7J@-B†PdÎj:yintimatµ%into£iæàŠ ¿ ‚ 9½ÖèkïÍ '[,‡/V1W5 q8½8˜@ AàA-BÀD†P¼Q  S-[Õ\ý\`ÎjBkëlo•ròsîx:yintre Bkòsintro Í !veyf W5Õ\eylü;eym Õ\ëleynëleyoëleytpÎjeytta¿ eyvŠ eyxÕ\eyxvdqr¿ eyxxxxxxÕ\eyyqrëlez¿ Ö"àAÕ\ÎjëlezbëlezgpÕ\ezhëlezhxÕ\ezjÕ\ezmëleznÕ\ezpëlezs ¿ Îjezw¿ ezyÎjezygjqÕ\fa[˜iŠ ¿ 9è"3(ˆ;J@.F†P¼Q-[Õ\1 ^dÎjBkëlòs:yfaaaaaaÕ\faal Õ\ëlfaanÕ\faapÕ\faarÕ\faasÕ\:yfigÏŠ ¿ £ 9Æ Ö¯"½8J@àA†P¼QÕ\BÜ_ jÎj Bkël$Ÿpòsîx:y7 S:ü;J@àA -BCN†P!Q‚Q¼QWV-[‰\Õ\+Ü_djÎjBkëlo9q:yuy indicatorÎjindicial ¿ indist¼Qindistinguishabl à¼Q îx individual†Pindstëlindstgëlinduc µ%¶JinduqÕ\ineÎjinear ¿ !vinearsol†PinedàC'†P inequalit†P¼QÎjinf¿ ':yinferr¼QÎjoinfgÕ\infiJ@àAÜ_Îj:yinfinJ@àA:yinfini7infinit‹xàŠ ¿ £ 9ûCkÍ µ%3(Ô+ ,[,Ë-‡/W578@J@˜@àA-BãM¼Q-[‰\Ü_dÎjÔmo s:yuàµ%¼QBk:yobvious ¿ j3(`+ƒ4½4R67½8S:ˆ; ü; <¥=˜@àA.F»I»M *O ‚Q¼QªY‰\Õ\ ^à^Ü_*d†ioÊo9qòs!vj Bkëlòsîx:yepsilongGGŠ ½k"'Ë-W5˜@¼QÕ\§bÎjBkëlîx:y epsilongfW5¼QÕ\Îjîx½4îibmintellinuxmaplinputcourimathtimehyperlinkcommoutputhelpheadnormaltextwarnerrorbulletitemfunctholonomdfinitaddcomputsystemequationsumdgfinitfunctionmulproductsagecallsequencdfddfdgngtgparameteriglistformgbgbgterordergroebnbasisynopsidescriptsatisfiedrespectiveallunderconsideratgenericsolutionslinearhomogeneouthesreprsentskewpolynomialcommonalgebraagoperatordefinthmustwithrespecttermorderoverstructurtablorealgebratermorddetailreturnquationinvolvpossibcrosderivatawarebaseanysolutspandimensionalvectorspacotherwisalgorithmcannotterminatfardfiniteonlyworkdifferentialshiftpartpackagcanuseddfiniafterperformcommandalwayaccesslongexamplnwithroebnexpmunubesseljbesseljexpgmugxgfnugfygfbesseljgdiffdxdynttdegfgggxgnuggbasihypergeomdygfdxgfgbasisgdxgygdygfcffafsyscollectdistributsysgmugffhffkfftffuffhnffenffinffufyffvffgnffsffjnffmf“$@½decalphaunixmaplinputcourimathtimehyperlinkcommoutputourihelpnormalimesheadoutputulletitemagfmomentsolvsolvsystemgeneratfunctequationassociatwithattributgrammarusagingsequenceqnsnumneweqnsparametersetultivariatnumbdifferentiatnonnegatintegneweqnoptionalequationattemptwillinfodescripttakemultivariatfunctiontrietheyeachrespectvariablasidsizeevaluatreturnunivariatthuscriticalimportancfirstmustmarkfunctionusefulfindaveraghighmomentmodificanusingthirdargumpresquationtoolonginteruptrecoverallowuserintervensommanualmanipulatspecialmethodrelisolutionobtainthessolvablfailcombstructspecificatcombstructinformathowwritcombstructgrammarcombinatoricagrammardefinitusesparticulardefinpropertgetfunctionexampleithgfagfeqnfollowuseuginternalnodeplantreeprodlabelledzgtgfexpggfeqnprodepsilonlabellugfegfgivenpossiblgeneratinunivariatturnithfurthcummulatbinaraÌ becaus/¿ Ƶ%'½4†PÜ_Îjîxuybeco¿ becom' {" ,J@†P¼QÎjBk!vbeeòsbefoobeforO"Š ¿ £ Æ{"q8ÀD¶J†P!Q¼QÜ_jBkëlòst:ybegi àAÎjbeginiË-W5Õ\ÎjëlbeginnÎjòst:ybeginniòsbehav¿  ,Bkbehavio Bk!vbehaviorý\behaviou£  behaviour#¿ £ ŒJ@Õ\òs!vîxbehind9bekëlbelgianbell+ài£ 9'ãM†P¼QÕ\!v ¿ £ P k￵%Ô+7ãM†P¼Q1V‰\Õ\*dÎjBkëlÔmo•r sòs:yexpansià¿ £ µ% expansiongÕ\expcount3w"˜]Š}expeBkexpec"expect3iŠ ¿ ï"V1½4†P¼QÕ\ÎjëlfcofgfJ@fcofzfJ@fcpŠ ¿ Õ\Îjfcpf ¿ ëlfcpfcpf¿ fcq ¿ Öëlfcr ¿ ëlfcrf ël:yfcs "Õ\fcsfgf7fct "Îjëlfctf "ël:yfcuël fcufdfqfiµ%fcv ¿ Õ\ëlfcw ¿ Õ\fcxÎjfcy Õ\Îjëlfcyf:yfcyfgsfhyfgpfiyfg:yfcyfk:yfcz Õ\ëlfd_ Š ¿ yÆÖè"µ%‡/J@àALD†P¼QÕ\/ ^à^ÎjBkëlHòs:yfda¼QfdalÕ\fdamÎjfdanÕ\fdaqÕ\fdarÕ\fdarfÕ\sgfe"sgg?sgk¿ sgnŠ sgramLDsgtcÎjsgwëlshŠ àAdÎjëlbme ÖW5bmffÕ\bmgfgëlbmisÕ\bmj"bmkëlbmm"bmmm"bmtf¿ bmvëlbmwiÕ\bmz¿ bmzuëlbn¿ Õ\ëlbncBkbnerì½8 ^•rbnfÕ\bnkàbnuëlbnzÕ\boŠ -B.F¼QÕ\ëlboardŠ bodÜ_bodyÜ_boeÕ\bold¿ cfnokrht¿ cfplBkcfpvt¿ cfrxÕ\cfsŠ cfsearŒcfsearch ŒãMcfsgŠ cfxÕ\cfzÎjcg´Š ë ¿ CÍ "'3(`+[,‡/W5/S:J@ AàA!Q‚Q¼QÕ\å]Ü_§bd*djÎj BkëloŸpIvîx:ykj à¿ kjhÕ\Bkëlo9q:yuyël•ròs!v Ivkwîx:y"uyòskw:ygaussian†Pgautoï{78tgautogï{78tgavëlgave'gax"gayëlgb;h¿ C?è3(‡/½4½8<†P Õ\ ^ëlkwgbasi3DC?è3(‡/ ½4Þ5½8ˆ;< ^ kwgbasisg ‡/½4gbbÕ\gbfÎjgbg/+C?è3( ‡/½4½8<†PëlkwgbifÕ\gbkÕ\gbmëlgbpg¿ gbs¿ gc# ¿ "¼QÕ\Îjëlòsgcbeëlgcd Í ˆ;ü;.FÕ\BkÊogcdeÍ gcdex$Í ˆ; ü;.F Êogcdexg Í ‡/gcdg Í Bkgceëlgcf"gchëlgciÕ\gclÕ\gclfÕ\gcnpëlgco¿ gcoeff†PgcrÎjÊo9qòs!vÆW5BkÞ5íibmintellinuxmaplinputcourimathtimehyperlinkcommoutputhelpheadnormalbulletitemfuncttypeclosedidealfinitdimensionalidealusagcallsequencggtgparametersetlistpolynomialtabldenottermorderalgebrasynopsidescriptcheckleadwithrespectgeneratzerodimensionagroebnbasiequivalthereforlessgeneralbutdoescomputanyopposinitexamplorenwithgroebnernapolynttermordtdegfgxgzgfygfalsupherenoeengbasiygftruegtruegalsotermordgroebntypeÌ^Weyl_algebra,Ore_algebraü;alcohols,autocombW5annihilators,Ore_algebraˆ;applyopr,Ore_algebraS:balls_and_urns,autocomb¼Qchannel_allocation,autocomb†Pcommutation_rules,Ore_algebra9qrlkÕ\rlknvëlrlll"rllllllqÕ\rlq ¿ "rlrkëlrlw¿ rlxëlrlyÎjrmCà¿ ‚ "‡/½8¥=àA¶JÕ\ ^ÎjBkëlòs:yrmaëlrmajxdëlrmalï78ÀDŠ}rmalf ^rmb ¿ ëlrmgÕ\rmhxÎjrmin¿ rminal Ü_§brminat½4 rministicŠ}rminiz½rmkÕ\rmlÕ\rmmmmÎjrmmmmmmÕ\rmnÕ\rmnbëlrmorÕ\rmord C‡/ rmqtkcuhohxëlrmsïq8Õ\cuss¿ fof/Š µ%ú+ <dÎjBkëlòst:y:ylfdÕ\lfdcmÕ\lfduÕ\lfe ¿ Õ\lfejlÕ\lfexf:ylffÕ\lffanÕ\lffcmÕ\lffeÕ\îx:yqpa¿ qpbrfÕ\qpdÕ\qpf ¿ Õ\qpgs¿ qpjfëlqpjfdëlqplëlqpo ,qpoch ,qpochhd qpochhamm3( , 7d qpochhammerg ,qpxÕ\qpzëlqq¿ ¼Q Õ\å]ëlqqbkëlqqcŠ qqdoÕ\qqdp¿ qqen¿ qqf Š "qqgå]qqgfå]qqhftj†Pqqiëlqqmtf¿ qqmxwëlqqnfÕ\•r:yphm¿ phnàphras:yphvëlphvzlŠ phy ¿ Õ\physic¿ ¼Q!vphysicaŠ physicalŠ ¿ !vphzvÕ\pi3&à¿ £ µ%W5†PÕ\ ÎjBkëlopiaëlpiajëlpibÕ\picëlpick †iBk {MPLTEXT 1 0 26 "hypergeom _Çcentur:yceo¼Qcertain3à¿ è"'Â;˜@”E¼QÎjîx:ycertaintÎjces Š ¿ cess è½4cestoròscevzoÕ\ceykkvëlcfcH¿ £ ½"µ%‡/7½8ˆ;J@àA.F†P¼QÕ\ ^djÎjBkël :ycfa:y cfaufkfcuf:ycfb¿ cfdÕ\cfe¿ cff"cfinòscfjÕ\cflBkcfnÎjcfnokrht¿ cfplBkcfpvt¿ cfrxÕ\cfsŠ cfsearŒcfsearch ŒãMcfsgŠ cfxÕ\cfzÎjcg“¶Š ë ¿ CÖÍ "'3(`+[,‡/W5/S:J@ AàA!Q‚Q¼QÕ\å]Ü_§bd*djÎj BkëloŸpIvîx:y†P-decalphaunixmaplinputcourimathtimehyperlinkcommoutputnormalextheadplotseatarrangemroblemphilippflajoletversjanuarngrowatuncheonettpeoplsitdownrandomtheyunfriendnevernextanothexpectnumberpersonoriginalproblemduefreedmansheppappearvolumsiamreviewvarioualternatformulattheminvolvfatmenneedmorestoolsimplifidescriptchanneloccupatmobiltelephoncentbelllabsconsecutradiostationarrivtrygrabfreebecauspossiblinterferencesnostatoccupalreadoccupiwhaproportnclearnumbliesomewherbetweenworksheetexplorthwaysolutsimilarmayfoundusinggfunpackagcommoschemaherewritimmediatspecificatuselisttorecisttorecprocedurguesrightdifferentiaequatexploitresultcapabilitintegrationasymptoticexpansbasicequationletggprobabilitgeneratfunctpgfumbercodebelowwetakeimplicitugvariablfirstindividualkgplusplushavebecomunavailablsubproblemsizenaturrandomnesassumpteachvaluwithequalnamelygfbal Õ\ëlfbaoÕ\fbapÕ\fbboÕ\fbbqÕ\fbclÕ\fbcnÕ\fbcpÕ\fbctÕ\fbdëlfbdlëlfbdpÕ\fbdqÕ\fbdvÕ\fbel Õ\ëlfbemÕ\fbeoÕ\fbeofÕ\fbepÕ\fbesÕ\fbfW;àŠ ¿ £ 9?"3(‡/W57ˆ;Â;ó=¼Q‰\Îjël:y fbfafcfafdfÎjfbfb:yfbfbfcf:yfbfc:yfbfcfˆ;fbfcfjfJ@fbfdfaf:yfbfjof ^fbflëlfbfmÕ\fbfofdfˆ;fbfpfˆ;fbfqÕ\fbfxÎjfbgmÕ\fbgnÕ\fbgoÕ\fbgpÕ\fbgtÕ\fbhqÕ\fbhs"fbhtÕ\fbil Õ\ëldimeŠ dimen‡/dimens#+Š ¿ C3(‡/ ^!v:ydimensio¿  dimensionaÞ5 dimensional'Š ¿ 3(‡/½4Þ5½8‚Qdimer Š !vdimergŠ dimergfŠ ding.FÎj9qdirÔ+directK#àŠ ¿ 95½ïÔ+Ë-J@˜@ÎjBkëlòs!v:ydirectl"directorP disW5disallow‡/ disappear-B!Q ^discardBk discordan:ydiscovP discoverK†PodiscretŠ àA•r!vdiscus !vdiscussàÕ\•r:ydisguisàAdisjoinÎjë J@jwi;¿ P ú+<‚QÕ\ ^Ì^à^Îjëlòs!v:y‰\ Ü_Bk 7ü;<$@J@ àA-B »M*O†P‚Q¼Q‰\Ü_jÎj Bk oÊoòskw:y<ó= ^ermin "µ%à<seat#†P!v:ysebëlsecµ% secfacmom‰\seco¼Qsecon 3(Bksecond6àiŠ ¿ £ ½'3(‡/½8X98@J@àA-B”E¶J¤N†P‚Q¼QXZ‰\Õ\)`ždBkoòs!vîx:ysecongÕ\sect+¿ £ µ%W5J@àAÕ\Îj9q:y section W5àAsedkå]îx sedgewick"seealãM9qtIvseealso‡aæ3i.ë ç ‚ û5CŒÆì:½?èkï¯K¿w"3(`+Ô+ú+ ,[,/è/Ð0V1ƒ4½4R67778½8X9S:ˆ;Â;ü;<¥=ó=$@8@˜@ A-BLDÀD”E.F»I¶J»MCN¤N*O!Q‚Q S1VWVªYXZ-[ý\˜]å] ^Ì^à^_Ü_)``§b|cd*d†iÔm}oÊoŸpÁq skwuyŠ}Õ\Š àAdÎjëldomiBkdominant àŠ ¿ ¼QÕ\Îjdominat¼Qdominiqu•rdominoŠ !v:ydominocŠ dominog:ydominohŠ dominovŠ don :y done# Š £ :KË-¼QÕ\Îjdont:ydontcar Îj:y dontcareg :y dontcaregf:y dontcaresgÎjdoronàAdoubl¿ 3(àA†Pòsdown Í †Pdozen†Pdp¿ ".FÕ\ëlòsdpg.FdpiÕ\dpj"dppëldprbëldpyëldpze¿ dpzgÕ\dqŠ .F†PÕ\ëllwwwwwwÕ\lx¿ W5¼QÕ\ A¼Qý\§bëllauricella ^lawà†P:ylar# Š ¿ P "' Ajîx:yroutin/Š µ%7àA‚Q ^ÎjBkoÊo:yrovid:y ,+ibmintellinuxmaplinputcourimathtimehyperlinkcommoutputourihelpheadnormallistitemulletfunctqbinomialqganalogubinomialcoefficientqfactorialfactorialqpochhammpochhammsymbolusagcallsequencqbinomialngmgxgparameteralgebraicexpresssynopsidescriptfunctionheusuqpochhammqpochhammergcomputymboldefingfqfactorialgcomputfactorialqfactorialqfactorialghammergngfqgfqbinomialgcoefficiqfactorialgexpandconvertqpochhammworkinvolvthesbehavliketheirclassicalcounterpartmeantsuchnfinitinfinitygmuslessmodulnumericalvaluproperpartmgfunpackaghowevyoumustloadmakethemavailablopleveltheycanalternativemadereadlireadlibrespectlybecomindependfuturversexamplwithnqpochhammagxgfseqrialrgvkqzgginfinitseealsoalsoymbolicmethodexposedgosaparticularweshallinterestenumeratverticalstripconstantwidthmultivariatgeneratingaveragpiecexpectproporttyperandomeasilywillalsoenablusestablishsequencupperlowerboundconnectivittaulimitinfinittauglimitggginfinitygtilenxnnxngcheesboardbeforgettstartneedloadlibrarycodedoingractioncombstrucgfunreadratasymptmplgfsolvreferenccohenetaliiphysicaxxifinfinchfavoritmathematicalhttpwwwmathsoftcomcgshlbatgourdonsalveffectasymptoticlinearrecurrenccoefficientdiscretematicvolguggenheimclarendonpreskenkenyonapproximatcoveringrocthacmstocrobertfilmoxygentungstenprocroyalsocietlondonsloasloanplouffencyclopediaintegacademicstepfirstobservtgcountislicknownexpresssincheightreachaddingverticalhavefibonaccicheckdirecttgruniondimerprodtgrgequenceguniongmonomergdimergzgprodgretrievcorrespondingfunctunlabellzgfmonomergfdimergftgfmorecaseexaminnowdrawÌhorizontallineatturnoutdocutencodmmlefmostverticaljusttouchtopmostsidepmleftmostintoheboundarwhilrightaddedassigneachcontainwordlengthalphabetfollowigdigitsplitdominoocatcolumnotherwissummarizourthereformpheightsbtwstandminuplusfeffgfefmfgfkfdfmflfkcolourgwhenevfindwoconsecutmsknowweththeytopsufficiincrementalbuildallpossiblconfiguratrecordstatufringthemcombinatotherdifferentstackhorizontalmonomerleftremaintransitsimilarruleorderharacterizordinatmarkbottommostwhoselevatgainputteverythtogethassociatsymbolhgvgsgheighyieldgrepsilonrodatomordinargfsyssysgpmgmpgfmmgfcgfhgfvgfsgfurthermorisolatcoefficiigfjgflgfchessboardxnxngfrespectivejgkglgsubsupindependentkindusederasmarkertaylorexpanionsericbtvuvogdoesappearsevalumatchthosgrammseqanothwaylargroughequatsatisfidiffeqtorecugrectoprocodevalffjlpvsvywmdprhzsrffcnlnvwxqdynpuhrbbkÍsntiockmobyyfuygiivueyvfqugzmtjnpndxldvgsngypogrvxkdagdxydkowgkenlfmhxoffctmymptoticestimatmuchfastseenelegantaccesthroughfullpartialfractdecompositdenominatorfpfconvertfullparfracfpfgsumgalphagrootofgcomecontributrootexpanselopelgsingularitmaincontributionsmallestmodulufsolvrootofnrootnrootdominantpoleclearcontributdfnextractpreviouexprssionproducesoperatorgarrowgffkbqkbdmhtfhdmchcyfgeneratingperformullomputsortincreasmodulisingularitkeyconsistdecidiesnumericalimplementoptionalargumlayersuserwanttakeaccountwritlayernbcfmungfnbcfsayerlayersgfefaffmffmfafcfsgsaqssecondapproximatioaccuratevalcqkpfzafnaddrescomputatmergtypesstijstijgiecegettotalderivatrespectsubstitutsstijsubdiffsstijgamplconfigurationtilltotqffifrfflfnbdpiecesnnbdpiecesngoopiecesasymptoticalequivalavnbdexpandavnbdgnribperasymptoccurrencsameirrelevantindeterminatpieceÎproportmgfkeptpieclocalforsubnbpforsubsnendendnbhnnbvnnbcnbhgnbvgnbcgzcplottroutinarchfigurabovweredominohndominovndominocdominovdominocscalcolorblueautomaticunctionhowautomattaskbinardoneallmpwordmpgrmpsallmpuencallstructmapcatpmmgmmpgmmmgppmgmppgmpmgpmpgbetweenletpatternpatterngsupposlettmeanputappldigitindependexcepthorizontalsufficrecursivelefbuiltmmpmmnreccomesfromstringidxprefixmullisttablprodrtoreindexulodreelseresultfitargetsubstrreccomesfromreccomesfromresulomeextraworkhereprinttablegmmpgfpmmgfepsilongppmgfmppgffmfjffjmffafentrsetgrammarfromtablatablalisttransiunimpmgfmmmgfftffcffjoatomgsolvusualgfsynmmmmmgfsysglfpmpgffbfffofcfaffbpfcfgfgprocedurreturngetgrammarnmptablnformptablnsetgrammarfromtablngetmmgfungrmmmmgfsyngrmnmmmgfsynsubscarriquitheavyvariatalleviatkeeponlygetmmgfunzgrmmsetgrammarfromtablnsubfsyssmallfroegbfhftftzkÏvcpboffgfiflklfoftdzfwffzffnffnbiggonesinherxgfresolutveryconsumbettalternatrunnresultfromasgfwfcyaunuubporevhugedzaiqcbedhoeabzjyqjajdpjexblspbpxkrlyzxrnoyergeckhtbuqgojhlzelyjlljdhmkqqfmabfvqnbqbgamfbbwzwkgrugvrqjapwndqjjqbejqqcrvhylzueyvhudofboqizkwdxqukkbxlydnabvqzqheoycegqffwhflnfuhaescaffoffspcqksgqdtontqihyftwrxqqeximtrjlmhsazbyiqhwphtumoqrckfiyyjajgqhzjcdffpqfxzeupcolqbktnbckecypojhocenlqzmwxisakawuispvwupnwxjyzqxzojqglakvnakvvafdqtrwrflwogvxtondtchaslhzimtffpqrhdgyzwgntuiilpsqnhpgltlnrksttksntnovoypmnpbybgdrqjbfsgjyqxemyvqghnatxozhhvnpnlspvlvqirulbdfyhorgmuhzvznnwsxqpqxgucxfrqstuqwhmwunsfrtzzspcxarrnfyshehhtthpghapglfofogjbjwomdecljphvzlsgnffnqwhghylzdjpwrksgyfzybfboffdsdenotapplicatnoknownalthoughapproximationagreentsigncoeffildexpftyobservatmultipltilphaithsingularityÐcompletchanglimimanymaximalminimizejointoverlaptakenalphagtherefoprovidtaobtainhavingpositdenomgzdsfrcylasttrickusetryguesrombergconvergencacceleratconvergrateformagmakesensduetooerroneouinitialaftersinglgetscloscommonacceptednvlxhfcfcfqqdfqfagfjfudzfxffenfgnfgnfvxrifrvyxurphdobcjahsghtioaplocpbfepfepfofimptfrfyhfenixfemwfgacswkbfphfshfverusbhnmpyofepfodfbofeofhogspegxuyutfcpfipoybfbqfeqckzrfepfepfhqfhqconclusvariouparameternumberprecisedefinerivfractioncoeficientabouttilingsaltoughpresentexponentialrowthdescribmputtheoremconnectvitsatisfnpbybgdrqjbfsgjyqxemyvqghnatxozhhvnpnlspvlvqirulbdfyhorgmuhzvznnwsxqpqxgucxfrqstuqwhmwunsfrtzzspcxarrnfyshehhtthpghapglfofogjbjwomdecljphvzlsgnffnqwhghylzdjpwrksgyfzybfboffdsdenotapplicatnoknownalthoughapproximationagreentsigncoeffildexpftyobservatmultipltilphaithsingularityÐBkÆ{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 " " 0 1 0 0 0 0 1 0 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 "" 18 262 "" 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 0 1 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 "" 18 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLEÒ "" -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 "" 18 270 "" 0 1 0 0 0 0 1 0 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 0 1 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 "" 18 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 1 18 0 0 0 0 0 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 "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 "HÓeading 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 O utput" 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 "" 3 256 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 }{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 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 23 "POLLARD'S RHO ALGORITHM "Ô }}{PARA 0 "" 0 "" {TEXT 280 0 "" }}{PARA 257 "" 0 "" {TEXT 256 11 "B runo Salvy" }}{PARA 258 "" 0 "" {TEXT -1 29 "(Version of January 27, 1 997)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "P ollard's " }{XPPEDIT 18 0 "rho" "I$rhoG6\"" }{TEXT -1 141 "-method is \+ an efficient technique used to find factors of integers. It is both ve ry efficient and very simple. We show in this worksheet how " } {HYPERLNK 17 "combstruct" 2 "combstruct" "" }{TEXT -1 5 " and " } {HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 162 " can be used to analyze a realistic combinatorial model of the algorithm and thus derive a pr obabilistic complexity analysis of this algorithm and variants of it. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(combstruct): with( gfun):" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 33 "Algorithm and combinat orial model" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 13 "The algorithm" }} {PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "N" "I\"NG6\"" }{TEXT -1 94 " is the integer of which a factoÕr is sought, the basic version \+ of the algorithm is as follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 29 "Pick up an arbitrary integer " }{XPPEDIT 18 0 "a" "I\"aG6\"" }{TEXT -1 5 " mod " }{XPPEDIT 18 0 "N" "I\"NG6\"" }{TEXT -1 6 ", set " }{XPPEDIT 18 0 "f(x)=x^2+a mod N" "/-%\"fG6#%\"xG -%$modG6$,&*$F&\"\"#\"\"\"%\"aGF-%\"NG" }}{PARA 0 "" 0 "" {TEXT -1 26 "Select at random a number " }{XPPEDIT 18 0 "x[0]" "&%\"xG6#\"\"!" } {TEXT -1 5 " mod " }{XPPEDIT 18 0 "N" "I\"NG6\"" }{TEXT -1 6 ", set " }{XPPEDIT 18 0 "y[0]=x[0]" "/&%\"yG6#\"\"!&%\"xG6#F&" }}{PARA 0 "" 0 " " {TEXT -1 8 "Iterate:" }}{PARA 0 "" 0 "" {TEXT -1 6 " " } {XPPEDIT 18 0 "i:=i+1; x[i]:=f(x[i-1]); y[i]:=f(f(y[i-1]))" "C%>%\"iG, &F$\"\"\"\"\"\"F&>&%\"xG6#F$-%\"fG6#&F*6#,&F$F&\"\"\"!\"\">&%\"yG6#F$- F-6#-F-6#&F66#,&F$F&\"\"\"F3" }}{PARA 0 "" 0 "" {TEXT -1 6 "until " } {XPPEDIT 18 0 "gcd(y[i]-x[i],N)<>1" "0-%$gcdG6$,&&%\"yG6#%\"iG\"\"\"&% \"xG6#F*!\"\"%\"NG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }Ö}{PARA 0 "" 0 "" {TEXT -1 151 "This is directly translated into \+ the following rough Maple procedure which returns a factor and the num ber of iterations performed to find this factor:" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 220 "pollard:=proc(N)\nlocal rnd, a, f, x, y, i, g ;\n rnd:=rand(N); a:=rnd(); x:=rnd(); y:=x;\n for i do\n \+ x:=x^2+a mod N; y:=(y^2+a mod N)^2+a mod N; g:=igcd(y-x,N);\n i f g<>1 then RETURN(g,i) fi\n od\nend: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Here are a few examples (Maple's " }{HYPERLNK 17 "nextpri me" 2 "nextprime" "" }{TEXT -1 67 " routine returns the smallest prime number following its argument)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "nextprime(900)*nextprime(20000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\")x*\\\"=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "pollard(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"$2*\"#J" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "pollard(\"\");" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$\"$2*\"#K" }}}{EXC×HG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "pollard(\"\"\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"$2*\"#I " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "nextprime(10^5)*nextpri me(10^6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"-4+I.+5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "pollard(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"(.++\"\"$<\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "pollard(\"\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"'.+5\"$O&" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "pollard(\"\"\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"'.+5\"$L#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "nextprime(10^6)*nextprime(10^7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"/d++\\++5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "pollard(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"(.++\"\"$#z" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "pollard(\"\");" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$\"(.++\"\"$!R" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "pollard(\"\"\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ \")>++5\"%q8Ø" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 23 "The combinatori al model" }}{PARA 0 "" 0 "" {TEXT -1 78 "The algorithm relies on the s tructure of the functional graph of the function " }{XPPEDIT 18 0 "f(x )=x^2+a mod p" "/-%\"fG6#%\"xG-%$modG6$,&*$F&\"\"#\"\"\"%\"aGF-%\"pG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 33 " is the smallest prime factor of " }{XPPEDIT 18 0 "N" "I\"NG6\"" }{TEXT -1 47 ". In this graph, the vertices are the integers " }{XPPEDIT 18 0 "0..p-1" ";\"\"!,&%\"pG\"\"\"\"\"\"!\"\"" }{TEXT -1 5 " mod " } {XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 57 " and the directed edges lin k each vertex to its image by " }{XPPEDIT 18 0 "f" "I\"fG6\"" }{TEXT -1 236 ". Since the number of points is finite, it is not difficult to see that the graph has the structure of a union of connected componen ts, each of these components being constituted of a cycle to which con verge trees. Since the polynomial " }{XPPEDIT 18 0 "f(x)" "-%\"fG6#%\" xG" }{TEXT -1 18 " has degree 2 and " }{XPPEDIÙT 18 0 "p" "I\"pG6\"" } {TEXT -1 36 " is prime, all the vertices (except " }{XPPEDIT 18 0 "a" "I\"aG6\"" }{TEXT -1 73 ") have in-degree 0 or 2, while they have out- degree 1. The prime factor " }{XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 66 " is assumed to be large, therefore the special case of the vertex \+ " }{XPPEDIT 18 0 "a" "I\"aG6\"" }{TEXT -1 125 " which has in-degree 1 \+ can be discarded as a first approximation. The combinatorial model is \+ thus expressed by the following " }{HYPERLNK 17 "combstruct grammar" 2 "combstruct[specification]" "" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 137 "G:=\{fungraph=Set(connected_component),\n \+ connected_component=Cycle(Prod(Z,bintree)),\n bintree=Union(Z,Prod( Z,Set(bintree,card=2)))\}:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 " T he execution of the algorithm is interpreted on this graph as follows: a random point " }{XPPEDIT 18 0 "x[0]" "&%\"xG6#\"\"!" }{TEXT -1 46 " of the graph is selected. Then two sequences " }{XPPEDIT 18 0 "xÚ[i]" "&%\"xG6#%\"iG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y[i]" "&%\"yG6#%\" iG" }{TEXT -1 32 " of iterates with initial value " }{XPPEDIT 18 0 "x[ 0]" "&%\"xG6#\"\"!" }{TEXT -1 374 " are computed, one of them proceedi ng one step at a time, while the other one proceeds by steps of length 2. Starting from a node of the graph, these two sequences eventually \+ reach a cycle, where they are bound to intersect. This is where the al gorithm stops.\n Under this model, the average number of steps requir ed by the algorithm is therefore related to two parameters: " }{TEXT 257 3 "(i)" }{TEXT -1 49 " the average distance from a point to its cy cle; " }{TEXT 258 4 "(ii)" }{TEXT -1 34 " the average length of the cy cles." }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 34 "Path length in planar binary trees" }}{PARA 0 "" 0 "" {TEXT -1 23 "We start with question \+ " }{TEXT 259 3 "(i)" }{TEXT -1 281 " above: the determination of the a verage distance from a point to its cycle. We first concentrate on a s imilar but simplerÛ problem, the computation of the average distance fr om a node to the root in a planar binary tree. This is related to a cl assical combinatorial parameter: the " }{TEXT 260 20 "internal path le ngth" }{TEXT -1 136 " of the tree, which is the sum of the distances f rom each of the nodes to the root.\nBinary trees are described by the \+ following grammar:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "bin:= \{bintree=Union(Epsilon,Prod(Z,bintree,bintree))\}:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "The counting sequence given by " }{HYPERLNK 17 "combstruct[count]" 2 "combstruct[count]" "" }{TEXT -1 50 " is constit uted by the well-known Catalan numbers:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "seq(count([bintree,bin,unlabelled],size=i),i=0..15); " }}{PARA 11 "" 1 "" {XPPMATH 20 "62\"\"\"F#\"\"#\"\"&\"#9\"#U\"$K\"\" $H%\"%I9\"%i[\"&'z;\"&'ye\"'7!3#\"'+Hu\"(SWn#\"(X[p*" }}}{PARA 0 "" 0 "" {TEXT -1 152 "The following Maple procedure takes as input a binary tree as produced by combstruct using the specÜification above and retu rns its internal path length:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "ipl:=proc(t)\n if type(t,name) then 0 # the tree is of the fo rm Z or Epsilon\n else # the tree is of the form Prod(Z,t1,t2)\n \+ ipl(op(2,t))+ipl(op(3,t))+size(t)-1\n fi\nend:\nsize:=proc(t) eval (subs([Prod=`+`,Z=1,Epsilon=0],t)) end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 218 "The internal path length is computed using a simple bije ction: each of the nodes on a branch from the root to a leaf are count ed once for each of the nodes below it. Here is an example on a tree o f size 5 generated by " }{HYPERLNK 17 "combstruct[draw]" 2 "combstruct [draw]" "" }{TEXT -1 30 " from the specification above:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "T:=draw([bintree,bin,unlabelled],si ze=5); ipl(T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG-%%ProdG6%%\"Z G-F&6%F(%(EpsilonG-F&6%F(-F&6%F(-F&6%F(F+F+F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 175 "We shall \+ compÝute the average internal path length by first computing the total \+ internal path length, i.e. the sum of the internal path lengths of all the binary trees of size " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 38 " and then dividing this number by the " }{XPPEDIT 18 0 "n" "I\"nG6 \"" }{TEXT -1 158 "th Catalan number. We first start by computing expe rimentally the first values of these numbers, generating all the trees of fixed sizes for small sizes with " }{HYPERLNK 17 "combstruct[allst ructs]" 2 "combstruct[allstructs]" "" }{TEXT -1 102 ", and computing t he sum of the internal path lengths of all these trees with the proced ure ipl above. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "for i to 5 do sum_ipl[i]:=`+`(op(map(ipl,allstructs([bintree,bin,unlabelled],s ize=i)))) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(sum_iplG6#\"\"\" \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(sum_iplG6#\"\"#F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(sum_iplG6#\"\"$\"#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%(sum_iplG6#Þ\"\"%\"#u" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(sum_iplG6#\"\"&\"$_$" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 36 "Combinatorial model for path lengths" }}{PARA 0 "" 0 "" {TEXT -1 290 "The numbers computed above can actually be derived more \+ efficiently, together with many other results related to path lengths \+ using combstruct's ability to deal with marks (atoms of size 0). The i dea consists in writing a combinatorial specification such that the nu mber of objects of size " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 81 " is exactly the sum of the internal path lengths of all the binary tre es of size " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 136 ". The combin atorial technique used here extends to other combinatorial structures \+ and leads to a combstruct-based analysis of Pollard's " }{XPPEDIT 18 0 "rho" "I$rhoG6\"" }{TEXT -1 437 "-algorithm.\nThe grammar for binary trees is modified to take into account \"decorated\" binary trees. A \+ binary tree is decorated by putting two marks (An for Ancestorß and De \+ for Descendant) on two nodes belonging to the same branch. The number \+ of ways of decorating a binary tree is then exactly its internal path \+ length. Couting the number of decorated trees therefore corresponds to summing the internal path lengths of all binary trees." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 390 "bin2:=\{bintree2=Prod(Z,\n \+ Union(leftright2,Prod(An,leftright1))),\n bintree1=Prod(Z,\n \+ Union(leftright1,Prod(De,bintree,bintree))),\n leftrig ht2=Union(Prod(bintree2,bintree),\n Prod(bintre e,bintree2)),\n leftright1=Union(Prod(bintree1,bintree),\n \+ Prod(bintree,bintree1)),\n An=Epsilon, De=Epsi lon\} union bin:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "The sequence \+ of cumulated internal path lengths is now derived in less than a secon d:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "seq(count([bintree2,b in2,unlabelled],size=i),i=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6, \"\"!\"\à"#\"#9\"#u\"$_$\"%)e\"\"%Yp\"&'yH\"'3g7\"'+z_" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Here are the 14 decorated trees of size 3 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "allstructs([bintree2,b in2,unlabelled],size=3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#70-%%ProdG 6$%\"ZG-F%6$%#AnG-F%6$%(EpsilonG-F%6$F'-F%6%%#DeGF--F%6%F'F-F--F%6$F'- F%6$F*-F%6$F.F--F%6$F'-F%6$F--F%6$F'-F%6$F*-F%6$F--F%6$F'-F%6%F2F-F--F %6$F'-F%6$F*-F%6$F3FE-F%6$F'-F%6$F*-F%6$F--F%6$F'-F%6$FEF--F%6$F'-F%6$ F--F%6$F'-F%6$F*FW-F%6$F'-F%6$F*-F%6$FEF3-F%6$F'-F%6$F*-F%6$-F%6$F'-F% 6%F2F3F-F--F%6$F'-F%6$F*-F%6$F--F%6$F'FC-F%6$F'-F%6$F*-F%6$F-Fgo-F%6$F '-F%6$FgnF--F%6$F'-F%6$F?F--F%6$F'-F%6$F*-F%6$FUF--F%6$F'-F%6$F*-F%6$F apF-" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 20 "Generating functions" } }{EXCHG {PARA 0 "" 0 "" {TEXT -1 151 "From the grammars describing bin ary trees and decorated binary trees, the average internal path length can be computed via generating functions. Using " }{HYPERLNK 17 "comb struct[gfsolve]" 2 "combstruct[gfsolve]á" "" }{TEXT -1 102 ", we first \+ derive the generating functions for the Catalan numbers and for the cu mulated path lengths:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "F: =subs(gfsolve(bin,unlabelled,z),bintree(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,$*&%\"zG!\"\",&\"\"\"F**$,&F*F*F'!\"%#F*\"\"#F(F *F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "S:=series(F,z,30);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"SG+in%\"zG\"\"\"\"\"!F'\"\"\"\" \"#\"\"#\"\"&\"\"$\"#9\"\"%\"#U\"\"&\"$K\"\"\"'\"$H%\"\"(\"%I9\"\")\"% i[\"\"*\"&'z;\"#5\"&'ye\"#6\"'7!3#\"#7\"'+Hu\"#8\"(SWn#\"#9\"(X[p*\"#: \")qwNN\"#;\"*!zW'H\"\"#<\"*+(QwZ\"#=\"+!>jsw\"\"#>\"+?/7kl\"#?\",?qEm W#\"#@\",SOc#[\"*\"#A\"-]OhfIM\"#B\".Ct9/**G\"\"#C\"._9SY>'[\"#D\"/_@2 `tO=\"#E\"//g\"4bL&p\"#F\"0g.v^zuj#\"#G-%\"OG6#F'\"#H" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Fpl:=subs(gfsolve(bin2,unlabelled,z ),bintree2(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$FplG,$*&,(*&%\"z G!\"\",&\"\"\"F,*$,&F,F,F)!\"%#F,\"\"#F*F,#F*F1#\"\"$F1F,F-F2F,,&F*F,F )\"â\"%F*F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Spl:=series(F pl,z,30);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$SplG+en%\"zG\"\"#\"\"# \"#9\"\"$\"#u\"\"%\"$_$\"\"&\"%)e\"\"\"'\"%Yp\"\"(\"&'yH\"\")\"'3g7\" \"*\"'+z_\"#5\"(!e&>#\"#6\"(s23*\"#7\")kGRP\"#8\"*OXV`\"\"#9\"*a*yxi\" #:\"+m9WiD\"#;\",/,MQ/\"\"#<\",O#[$\\C%\"#=\"-C>kwB<\"#>\"-c@G+\"*p\"# ?\".C=U0A$G\"#@\"/O0GaGY6\"#A\"/k@ArXNY\"#B\"0/g&R(>J(=\"#C\"0sAL!)>Rc (\"#D\"1)[qYtyD0$\"#E\"2c'pVLZEJ7\"#F\"2Ot9R\"=!R'\\\"#G-%\"OG6#\"\"\" \"#H" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "The average path length i s simply the ratio of these coefficients:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "seq(coeff(Spl,z,i)/coeff(S,z,i),i=0..28);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6?\"\"!F#\"\"\"#\"#9\"\"&#\"#P\"\"(#\"$w\"\"#@ #\"$(R\"#L#\"%Yp\"$H%#\"&$*[\"\"$:(#\"&/I'\"%JC#\"'v>8\"%*>%#\"(!z(4\" \"&$RH#\"($>qA\"&.?&#\"(;#[$*\"'Dd=#\")<$z\">\"'0VL#\"*a*yxi\"(X[p*#\" +L2A\"G\"\")N)yw\"#\"+_+<>_\")&RA['#\",fqL71\"\"*v'4%>\"#\",i4K)=')\"* &fJO))#\"-R02vZ<\"+0,.T;#\"-caNã^!3(\"+bnc;h#\".n]ycGV\"\",b/KN9\"#\"/# 36cGxJ#\"-Do!)H:<#\"/,!*[$*z#o%\"-Jo.wCK#\"0oI3&*z4*=\".j.g'[:7#\"06Q$ =Ms:Q\".>S8>fH##\"19CfL=;yI\"/,!HxQ$Q<#\"1 " 0 "" {MPLTEXT 1 0 11 "evalf([\"]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7?\"\"!F$$\"\"\"F$$\"+++++G!\"*$\"+'G9dG&F)$\"+\"Q _4Q)F)$\"+....7!\")$\"+>U6>;F0$\"+jq$H3#F0$\"+i1p\"f#F0$\"+T-,VJF0$\"+ Y)o[t$F0$\"+8R]lVF0$\"+$QkL.&F0$\"+O:2PdF0$\"+#G*QvkF0$\"+[6?ZsF0$\"+h @\\^!)F0$\"+YVL()))F0$\"+p\\(Qv*F0$\"+*oK]1\"!\"($\"+cgfd6FM$\"+N(4ID \"FM$\"+%Q57N\"FM$\"+1(Q@X\"FM$\"+N!Rdb\"FM$\"+y*e>m\"FM$\"+4\"\\2x\"F M$\"+(pi?)=FM" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 167 "Empirically, th ese numbers grow slightly faster than linearly with the size of the tr ee. A closed-form for the average path length can be established rigor ously using " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 112 ". The ge nerating functions being algebraic, they satisfy linear differential e quations. These can be derived by " }{HYPERLNK 17 "gfun[holeäxprtodiffe q]" 2 "gfun[holexprtodiffeq]" "" }{TEXT -1 109 ". From these different ial equations, a linear recurrence satisfied by the Taylor coefficient s is obtained by " }{HYPERLNK 17 "gfun[diffeqtorec]" 2 "gfun[diffeqtor ec]" "" }{TEXT -1 74 ". It turns out that these recurrences fall into \+ a class for which Maple's " }{HYPERLNK 17 "rsolve" 2 "rsolve" "" } {TEXT -1 60 " can find a solution.\n Here are the differential equati ons:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "deqF:=holexprtodiff eq(F,y(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%deqFG,(\"\"\"F&*&,&! \"\"F&%\"zG\"\"#F&-%\"yG6#F*F&F&*&,&F*F)*$F*F+\"\"%F&-%%diffG6$F,F*F&F &" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "deqFpl:=holexprtodiffe q(Fpl,y(z));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'deqFplG<&/---%#@@G6 $%\"DG\"\"#6#%\"yG6#\"\"!\"\"%/-F/F0F1/--F,F.F0F1,(*$%\"zGF-\"\"'*&,** $F:\"\"$\"\")F9!#EF:\"#5!\"\"\"\"\"FD-F/6#F:FDFD*&,**$F:F2\"#;F>!#CF9 \"\"*F:FCFD-%%diffG6$FEF:FDFD" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 " From there, the reåcurrences follow:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "recF:=diffeqtorec(deqF,y(z),u(n));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%recFG<$/-%\"uG6#\"\"!\"\"\",&*&,&\"\"#F+%\"nG\"\"% F+-F(6#F0F+F+*&,&!\"#F+F0!\"\"F+-F(6#,&F0F+F+F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "recFpl:=diffeqtorec(deqFpl,y(z),u(n));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'recFplG<&,**&,&\"\")\"\"\"%\"nG\"# ;F*-%\"uG6#F+F*F**&,&!#]F*F+!#CF*-F.6#,&F+F*F*F*F*F**&,&\"#GF*F+\"\"*F *-F.6#,&F+F*\"\"#F*F*F**&,&!\"%F*F+!\"\"F*-F.6#,&F+F*\"\"$F*F*F*/-F.6# F>F>/-F.6#\"\"!FM/-F.6#F*FM" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "T he recurrence satisfied by the Catalan numbers being linear of order 1 , it is obvious that Maple will be able to solve it:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Cat:=rsolve(recF,u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$CatG**)\"\"%%\"nG\"\"\"-%&GAMMAG6#,&F(F)#F)\"\"#F )F)-F+6#,&F(F)F/F)!\"\"%#PiG#F3F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 261 "It is however more surprising that rsolve canæ compute the solutio n of the 3rd order recurrence satisfied by the cumulated path lengths. This is due to the implementation of the recent algorithm by M. Petko vsek for hypergeometric solutions of linear recurrences." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Pl:=rsolve(recFpl,u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#PlG,&*0)\"\"%%\"nG\"\"\",&F)F*#F*\"\"$F*F *)#F*\"\"#F)F*)F0F)F*-%&GAMMAG6#,&F)F*F/F*F*-F36#,&F)F*F0F*!\"\"%#PiG# F9F0!\"$F'F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "After some cleani ng up, we have thus proved the following." }}}{EXCHG {PARA 259 "" 0 " " {TEXT 264 11 "Proposition" }{TEXT -1 2 ". " }{TEXT 261 83 "The avera ge internal path length in a random planar unlabelled binary tree of s ize " }{XPPEDIT 262 0 "n" "I\"nG6\"" }{TEXT 263 29 " under the uniform model is \n" }{XPPEDIT 18 0 "4^n*(n+1)/binomial(2*n,n)-3*n-1" ",(*() \"\"%%\"nG\"\"\",&F&F'\"\"\"F'F'-%)binomialG6$*&\"\"#F'F&F'F&!\"\"F'*& \"\"$F'F&F'F/\"\"\"F/" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 ç45 "seq(4^i*(i+1)/binomial(2*i,i)-3*i-1,i=1..10);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"!\"\"\"#\"#9\"\"&#\"#P\"\"(#\"$w\" \"#@#\"$(R\"#L#\"%Yp\"$H%#\"&$*[\"\"$:(#\"&/I'\"%JC#\"'v>8\"%*>%" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "From there, the asymptotic behavio ur is well within the reach of Maple's asympt command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "map(combine,asympt(Pl/Cat,n),exp); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*&%#PiG#\"\"\"\"\"#*$%\"nG!\"\"# !\"$F(F'F*F-*&F%F&F)#F+F(#\"\"*\"\")F+F'*&F%F&F)F&#\"#<\"$G\"*&F%F&F)# \"\"$F(#F9\"%C5-%\"OG6#*$F)#\"\"&F(F'" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 40 "Average distance from a point to a cycle" }}{PARA 0 "" 0 "" {TEXT -1 46 "We now come back to the analysis of Pollard's " } {XPPEDIT 18 0 "rho" "I$rhoG6\"" }{TEXT -1 209 "-algorithm. The analysi s of the average distance from a node to its cycle proceeds exactly as in the case of binary trees by decorating the corresponding trees. Th e grammar is derived from the grammar G above:" }}{EXCHGè {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "G;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<%/%)fungra phG-%$SetG6#%4connected_componentG/F)-%&CycleG6#-%%ProdG6$%\"ZG%(bintr eeG/F2-%&UnionG6$F1-F/6$F1-F'6$F2/%%cardG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 176 "The process of decoration consists in isolating the path between ancestor and descendant in the combinatorial structure. \+ For instance, non-planar binary trees are described by " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "npbin:=\{bintree=subs(G,bintree)\}; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&npbinG<#/%(bintreeG-%&UnionG6$% \"ZG-%%ProdG6$F+-%$SetG6$F'/%%cardG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 335 "Here, Set indicates that the respective position of the \+ two subtrees does not count. In the labelled case, we can therefore de cide that the decorated branch will always be the leftmost one (instea d of considering all the paths leftright1 and leftright2 as in the cas e of planar binary trees). Thus the decorated grammar in this case is " }}}{EXCHG {PéARA 0 "> " 0 "" {MPLTEXT 1 0 277 "bin3:=\{bintree2=Prod( Z,Union(Prod(bintree2,bintree),\n Prod(An, bintree1,bintree))),\n bintree1=\n Prod(Z,Union(Prod(b intree1,bintree),\n Prod(De,Set(bintree,card=2) ),De)),\n An=Epsilon,De=Epsilon\} union npbin:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 149 "Again, we can check that the first values coin cide with the result produced by computing the internal path length on all the non-planar binary trees:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "ipl:=proc(t) if type(t,name) then 0 else size(t)-1+i pl(op([2,1],t))+ipl(op([2,2],t)) fi end:\nsize:=proc(t) local i;eval(s ubs([Prod=`+`,Set=`+`,seq(i=1,i=indets(t,name))],t)) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "for i to 6 do i,`+`(op(map(ipl,alls tructs([bintree,npbin,labelled],size=i)))) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"#\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"$\"\"'" }ê}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"&\"$g $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"'\"\"!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 52 "seq(count([bintree2,bin3,labelled],size=i),i=1 ..11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-\"\"!F#\"\"'F#\"$g$F#\"&Sl$F #\"(+o*eF#\"++w2-9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 159 "The same p rocess readily extends to the functional graphs involved in Pollard's \+ algorithm by a decomposition of sets and cycles which isolates the mar ked part." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 296 "G1:=\{fungrap h=Prod(connected_component1,\n Set(connected_compone nt)),\n connected_component1=Prod(Z,\n Union(Prod(An,bintr ee1),bintree2),\n Sequence(Prod(Z,bintree))),\n conn ected_component=Cycle(Prod(Z,bintree))\}\nunion bin3: \+ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 229 "We first write pr ocedures to compute the distance from nodes to their cycles in the non -ëdecorated graphs so that we can check on the first few values that ou r grammar for decorated graphs is consistent with the non-decorated on e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 351 "iplfg:=proc(g) `+`(o p(map(iplcc,g))) end:\niplcc:=proc(cc) `+`(op(map(iplbt,cc))) end:\nip lbt:=proc(bt) if type(bt,name) then 0 else size(op(2,bt))+iplt(op(2,bt )) fi end:\niplt:=proc(t) if type(t,name) then 0 else size(t)-1+iplt(o p([2,1],t))+iplt(op([2,2],t)) fi end:\nsize:=proc(t) local i; eval(sub s([Set=`+`,Prod=`+`,seq(i=1,i=indets(t,name))],t)) end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Here are a few tests:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "to 4 do t:=draw([fungraph,G,labelled],size= 6); print(t,iplfg(t)) od:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%$SetG6$ -%&CycleG6#-%%ProdG6$&%\"ZG6#\"\"\"&F-6#\"\"&-F'6#-F*6$&F-6#\"\"$-F*6$ &F-6#\"\"%-F$6$&F-6#\"\"#&F-6#\"\"'FF" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%$SetG6#-%&CycleG6$-%%ProdG6$&%\"ZG6#\"\"%&F-6#\"\"\"-F*6$&F-6#\" \"'-F*6$&F-6#\"\"&-F$6$&F-6#\"\ì"#&F-6#\"\"$F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%$SetG6$-%&CycleG6#-%%ProdG6$&%\"ZG6#\"\"&&F-6#\"\"\"- F'6#-F*6$&F-6#\"\"%-F*6$&F-6#\"\"'-F$6$&F-6#\"\"#&F-6#\"\"$F>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$-%$SetG6#-%&CycleG6#-%%ProdG6$&%\"ZG6# \"\"&-F*6$&F-6#\"\"%-F$6$&F-6#\"\"\"-F*6$&F-6#\"\"'-F$6$&F-6#\"\"#&F-6 #\"\"$\"#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "The number of funct ional graphs of fixed size grows quite fast:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "seq(count([fungraph,G,labelled],size=i),i=1..10) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"!\"\"#F#\"#OF#\"%+=F#\"'+k " 0 "" {MPLTEXT 1 0 52 "map(iplfg,allstructs([fungra ph,G,labelled],size=2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"\"F$ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "sort(map(iplfg,allstruc ts([fungraph,G,labelled],size=4)));" }}{PARA 11 "" 1 "" {XPPMATH í20 "6 #7F\"\"#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%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "convert(\",` +`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$3\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "On the other hand, here is the counting sequence for decorated graphs:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "seq(c ount([fungraph,G1,labelled],size=i),i=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"!\"\"#F#\"$3\"F#\"&S/\"F#\"(+!Q;F#\"*+[Q#Q" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "It takes 11 minutes to check that 10440 is also the value we get by generating all the binary functiona l graphs of size 6." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "We now com pute the generating functions with " }{HYPERLNK 17 "combstruct[gfsolve ]" 2 "combstruct[gfsolve]" "" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "F:=subs(gfsolve(G,labelled,z),fungraph(z));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG*$,&*$%\"zG\"\"#!\"#\"\"\"F+#!\" \"F)" }}}{EXCHG {PARA 0 "> " 0 î"" {MPLTEXT 1 0 46 "Fpl:=subs(gfsolve(G 1,labelled,z),fungraph(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$FplG **%\"zG\"\"\",&*&F&!\"\",&\"\"#F'*$,&*$F&F,!\"#F'F'#F'F,F0F'#F*F,F&F,F ',(F'F'F/!\"%*$F&\"\"%F6F*F.F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Again, we obtain closed-forms from these generating functions by appl ying " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 110 " to find the di fferential equation they satisfy and from there the recurrence satisfi ed by their coefficients:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "deqF:=holexprtodiffeq(F,y(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%%deqFG<$,&*&%\"zG\"\"\"-%\"yG6#F(F)\"\"#*&,&!\"\"F)*$F(F-F-F)-%%diff G6$F*F(F)F)/-F+6#\"\"!F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "deqFpl:=holexprtodiffeq(Fpl,y(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%'deqFplG<%/-%\"yG6#\"\"!F*/--%\"DG6#F(F)F*,(*&,(*$%\"zG\"\"&\"#C*$F 4\"\"$!#CF4\"\"'\"\"\"-F(6#F4F;F;*&,**$F4\"\"#F:!\"\"F;*$F4F:\"\")*$F4 \"\"%!#7F;-%%diffG6$F " 0 ï"" {MPLTEXT 1 0 34 "recF:=diffeqtorec(deqF,y(z),u(n));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%recFG<%/-%\"uG6#\"\"!\"\"\",&*&,&%\"nG\"\"#F0F+F+- F(6#F/F+F+*&,&!\"#F+F/!\"\"F+-F(6#,&F/F+F0F+F+F+/-F(6#F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "recFpl:=diffeqtorec(deqFpl,y(z),u(n ));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'recFplG<)/-%\"uG6#\"\"!F*/-F (6#\"\"\"F*/-F(6#\"\"$F*/-F(6#\"\"&F*/-F(6#\"\"#F./-F(6#\"\"%#\"\"*F:, **&,&\"#CF.%\"nG\"\")F.-F(6#FEF.F.*&,&!#[F.FE!#7F.-F(6#,&FEF.F:F.F.F.* &,&FE\"\"'\"#IF.F.-F(6#,&FEF.F>F.F.F.*&,&FE!\"\"!\"'F.F.-F(6#,&FEF.FRF .F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "Unfortunately, due to a weakness in Maple's current version of rsolve, the solutions to these recurrences are not found" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "rsolve(recF,u(n)),rsolve(recFpl,u(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$-%'rsolveG6$<%/-%\"uG6#\"\"!\"\"\",&*&,&%\"nG\"\"#F1F,F ,-F)6#F0F,F,*&,&!\"#F,F0!\"\"F,-F)6#,&F0F,F1F,F,F,/-F)6#F,F+F2-F$6$<)/ F(F+F;/-F)6#\"\"$F+/-F)6#\"\"&ðF+/-F)6#F1F,/-F)6#\"\"%#\"\"*F1,**&,&\"# CF,F0\"\")F,F2F,F,*&,&!#[F,F0!#7F,F8F,F,*&,&F0\"\"'\"#IF,F,-F)6#,&F0F, FPF,F,F,*&,&F0F7!\"'F,F,-F)6#,&F0F,FhnF,F,F,F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "However, Maple can find the solution if we help it \+ by taking into account the parity of the generating functions:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "deqF2:=holexprtodiffeq(subs( z=z^(1/2),F),y(z)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "deqF pl2:=holexprtodiffeq(subs(z=z^(1/2),normal(Fpl)),y(z)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "recF2:=diffeqtorec(deqF2,y(z),u(n)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&recF2G<$/-%\"uG6#\"\"!\"\"\",& *&,&F+F+%\"nG\"\"#F+-F(6#F/F+F+*&,&F/!\"\"F5F+F+-F(6#,&F/F+F+F+F+F+" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "recFpl2:=diffeqtorec(deqFp l2,y(z),u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(recFpl2G<&/-%\"uG 6#\"\"!F*,**&,&\"#7\"\"\"%\"nG\"\")F/-F(6#F0F/F/*&,&F0!#7!#CF/F/-F(6#, &F0F/F/F/F/F/*&,&\"#:F/F0\"\"'F/-F(6#,&F0F/\"\"#F/F/F/*&,&F0!ñ\"\"!\"$F /F/-F(6#,&F0F/\"\"$F/F/F//-F(6#F/F//-F(6#FB#\"\"*FB" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "nb_fg:=subs(n=n/2,rsolve(recF2,u(n)));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&nb_fgG**)\"\"#,$%\"nG#\"\"\"F'F+-% &GAMMAG6#,&F*F+F)F*F+-F-6#,&F+F+F)F*!\"\"%#PiG#F3F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "tot_pl:=map(simplify,subs(n=n/2,rsolve(re cFpl2,u(n))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'tot_plG,()\"\"#,$ %\"nG#\"\"\"F'F+*&)F',&F)F*!\"\"F+F+F)F+F+**)F',&F+F+F)F*F+-%&GAMMAG6# ,&#\"\"$F'F+F)F*F+-F46#F2F/%#PiG#F/F'F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "This is summarized by:" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 265 11 "Proposition" }{TEXT -1 2 ": " }{TEXT 266 100 "The averag e distance from a point to a cycle in a random binary non-planar funct ional graph of size " }{XPPEDIT 267 0 "n" "I\"nG6\"" }{TEXT 268 13 " i s given by " }{XPPEDIT 18 0 "sqrt(Pi)*GAMMA(n/2+2)/n/GAMMA(n/2+1/2)-1- 1/n" ",(**-%%sqrtG6#%#PiG\"\"\"-%&GAMMAG6#,&*&%\"nGF(\"\"#!\"\"F(\"\"# F(F(F.F0-F*6#,&*&F.F(\"\"#òF0F(*&\"\"\"F(\"\"#F0F(F0F(\"\"\"F0*&\"\"\"F (F.F0F0" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "From t here a direct asymptotic expansion follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "map(simplify,asympt(subs(tot_pl/nb_fg/n),n));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,2*(\"\"##\"\"\"F%%#PiGF&*$%\"nG!\"\"# F+F%#F'\"\"%F+F'*(F%F&F(F&F)F&#\"\"*\"#;F)F+**F%F&F(F&F*F+F)F&#\"#<\"$ G\"**F%F&F(F&F*!\"#F)F&#\"\"$\"$7&**F%F&F(F&F*!\"$F)F&#!$\"=\"%#>)-%\" OG6#*&F*!\"%F)F&F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "as_co st_1:=op(1,\"):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Thus under our probabilistic model, the first stage of Pollard's " }{XPPEDIT 18 0 "r ho" "I$rhoG6\"" }{TEXT -1 32 "-algorithm takes asymptotically " } {XPPEDIT 18 0 "C*sqrt(p)" "*&%\"CG\"\"\"-%%sqrtG6#%\"pGF$" }{TEXT -1 14 " steps, where " }{XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 63 " is th e smallest prime factor of the number to be factored and " }{XPPEDIT 18 0 "C=sqrt(2*Pi)/4" "/%\"CG*&-%%sqrtG6#*&\"\"#\"\"\"%#PiGFó*F*\"\"%! \"\"" }{TEXT -1 1 "." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 28 "Average length of the cycles" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "After bot h sequences (" }{XPPEDIT 18 0 "x[i]" "&%\"xG6#%\"iG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y[i]" "&%\"yG6#%\"iG" }{TEXT -1 438 ") have reached the cycle, the number of steps before the end of Pollard's algorithm \+ is bounded by the length of this cycle. Since this length might be cor related to the number of steps before, it is not a priori sufficient t o compute the average length of the cycles in a random graph obeying o ur grammar. Instead, we modify the decorated graphs so that the Ancest or is now an element of the cycle. The number of decorated graphs of s ize " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 96 " is thus the sum for all the nodes of the length of their cycle, summed over all graphs of size " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 26 ". Here is the new \+ grammar:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 247 "G2:=remove(typ e,G1,identicôal(connected_component1)=anything)\nunion \{connected_comp onent1=\n Prod(Z,\n Union(bintree1,Prod(De,bintree)),\n \+ Sequence(Prod(Z,bintree)),\n Prod(Z,An,bintree),\n \+ Sequence(Prod(Z,bintree)))\}:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "The corresponding generating function is again algebraic:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Fplcycle:=factor(normal(subs (gfsolve(G2,labelled,z),fungraph(z))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)FplcycleG,$*(%\"zG\"\"#,&!\"\"\"\"\"*$,&*$F'F(!\"#F+F+#F+F(F+ F+,&F*F+F.F(F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "series( Fplcycle,z,11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+-%\"zG\"\"#\"\"%\" \"*\"\"'\"#H\"\")#\"$D$\"\"%\"#5-%\"OG6#\"\"\"\"#7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Again, a closed form follows for the coefficients :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "holexprtodiffeq(subs(z =z^(1/2),Fplcycle)/z,y(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$,(\" \"#\"\"\"*&,(\"\"$F&*$%\"zGF%\"#7F+!#7F&-%õ\"yG6#F+F&F&*&,*F+\"\"'!\"\" F&*$F+F)\"\")F*F-F&-%%diffG6$F.F+F&F&/-F/6#\"\"!F=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "diffeqtorec(\",y(z),u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/-%\"uG6#\"\"!F(/-F&6#\"\"\"\"\"#/-F&6#F-\"\"*,* *&,&\"#7F,%\"nG\"\")F,-F&6#F6F,F,*&,&F6!#7!#CF,F,-F&6#,&F6F,F,F,F,F,*& ,&\"#:F,F6\"\"'F,-F&6#,&F6F,F-F,F,F,*&,&F6!\"\"!\"$F,F,-F&6#,&F6F,\"\" $F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "rsolve(\",u(n)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,()\"\"#%\"nGF%*,F$F%)#\"\"\"F%F&F *-%&GAMMAG6#,&#\"\"$F%F*F&F*F*-F,6#,&F&F*F*F*!\"\"%#PiG#F4F%!\"%*&F$F* F&F*F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "T1:=map(simplify, subs(n=n/2-1,\")+tot_pl);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T1G,,) \"\"#,$%\"nG#\"\"\"F'F'**)F',&F+F+F)F*F+-%&GAMMAG6#,&F*F+F)F*F+-F06#F( !\"\"%#PiG#F5F'F5*&)F',&F)F*F5F+F+,&F)F+!\"#F+F+F+*&F9F+F)F+F+**F-F+-F 06#,&#\"\"$F'F+F)F*F+-F06#F.F5F6F7F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "We thus get the following conclusion:" }}}{EXCHG {PARA 261 "" ö 0 "" {TEXT 272 7 "Theorem" }{TEXT -1 2 ". " }{TEXT 269 41 "The average number of steps of Pollard's " }{XPPEDIT 270 0 "rho" "I$rhoG6\"" } {TEXT 271 59 "-algorithm under the probabilistic model is bounded betw een" }{XPPEDIT 18 0 "sqrt(Pi)*GAMMA(n/2+2)/n/GAMMA(n/2+1/2)-1-1/n" ",( **-%%sqrtG6#%#PiG\"\"\"-%&GAMMAG6#,&*&%\"nGF(\"\"#!\"\"F(\"\"#F(F(F.F0 -F*6#,&*&F.F(\"\"#F0F(*&\"\"\"F(\"\"#F0F(F0F(\"\"\"F0*&\"\"\"F(F.F0F0 " }{TEXT -1 4 " " }{TEXT 273 3 "and" }{TEXT -1 3 " " }{XPPEDIT 18 0 "2*sqrt(Pi)/n*GAMMA(n/2+1)/GAMMA(n/2+1/2)-2-1/n" ",(*,\"\"#\"\"\" -%%sqrtG6#%#PiGF%%\"nG!\"\"-%&GAMMAG6#,&*&F*F%\"\"#F+F%\"\"\"F%F%-F-6# ,&*&F*F%\"\"#F+F%*&\"\"\"F%\"\"#F+F%F+F%\"\"#F+*&\"\"\"F%F*F+F+" } {TEXT -1 2 ",\n" }{TEXT 274 6 "where " }{XPPEDIT 275 0 "n" "I\"nG6\"" }{TEXT 276 137 " is the smallest prime factor of the integer whose fac torization is sought. \nAsymptotically, this number of steps therefore behaves like " }{XPPEDIT 18 0 "C*sqrt(n)" "*&%\"CG\"\"\"-%%sqrtG6#%\" nGF$" }{TEXT -1 2 ", " }{TEXT 277 5÷ "with " }{XPPEDIT 278 0 "C" "I\"CG 6\"" }{TEXT 279 3 " in" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(2*Pi)/4.. sqrt(2*Pi)/2" ";*&-%%sqrtG6#*&\"\"#\"\"\"%#PiGF)F)\"\"%!\"\"*&-F%6#*& \"\"#F)F*F)F)\"\"#F," }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Here is the verification of the second part:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "map(simplify,asympt(T1/nb_fg/n,n,3));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(\"\"##\"\"\"F%%#PiGF&*$%\"nG!\"\"# F+F%F&!\"#F'*(F%F&F(F&F)F&#\"\"&\"\")F)F+-%\"OG6#*&F*F+F)F&F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "as_cost_2:=op(1,\"):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(\"\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$*$%\"nG!\"\"#F'\"\"#$\"+PTJ`7!\"*$!\"#\"\"!\" \"\"*$F%#F0F)$\"+rEkm:F,F%$F'F/-%\"OG6#*&F&F'F%F2F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Since the smallest prime factor of a number " } {XPPEDIT 18 0 "N" "I\"NG6\"" }{TEXT -1 13 " is of order " }{XPPEDIT 18 0 "O(sqrt(N))" "-%\"OG6#-%%sqrtG6#%\"NG" }{TEXT -1 95 ", it follows thøat the (arithmetic) complexity of Pollard's algorithm under this mo del grows in " }{XPPEDIT 18 0 "O(N^(1/4))" "-%\"OG6#)%\"NG*&\"\"\"\"\" \"\"\"%!\"\"" }{TEXT -1 1 "." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 33 "Extension: polynomials of degree " }{XPPEDIT 18 0 "m" "I\"mG6\"" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "When the factor " } {XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 42 " to be found is known a pri ori to satisfy " }{XPPEDIT 18 0 "p=1 mod m" "/%\"pG-%$modG6$\"\"\"%\"m G" }{TEXT -1 10 " for some " }{XPPEDIT 18 0 "m" "I\"mG6\"" }{TEXT -1 110 " known in advance, Pollard conjectured that the number of steps i n his algorithm could be reduced by a factor " }{XPPEDIT 18 0 "sqrt(m- 1)" "-%%sqrtG6#,&%\"mG\"\"\"\"\"\"!\"\"" }{TEXT -1 25 " by using the p olynomial " }{XPPEDIT 18 0 "f(x)=x^m+a mod N" "/-%\"fG6#%\"xG-%$modG6$ ,&)F&%\"mG\"\"\"%\"aGF-%\"NG" }{TEXT -1 12 " instead of " }{XPPEDIT 18 0 "f(x)=(x^2+a mod N" "/-%\"fG6#%\"xG-%$modG6$,&*$F&\"\"#\"\"\"%\"a GF-%\"NG" }{TEXT -1 98 ". Brent and Pollaùrd used this idea to factor t he 8th Fermat number, whose factors are known to be " }{XPPEDIT 18 0 " 1 mod 2^(8+2)" "-%$modG6$\"\"\")\"\"#,&\"\")\"\"\"\"\"#F*" }{TEXT -1 63 ". We know consider Pollard's under the probabilistic model for " } {XPPEDIT 18 0 "m=6" "/%\"mG\"\"'" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Here is the grammar for functional graphs with " } {XPPEDIT 18 0 "m" "I\"mG6\"" }{TEXT -1 16 " as a parameter:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "Gm:=\{fg=Set(cc),cc=Cycle(Pr od(Z,Set(tree,card=m-1))),\n tree=Union(Z,Prod(Z,Set(tree,card=m))) \}:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Here is the corresponding \+ decorated grammar for the distances to the cycles:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 402 "Gm2:=\{fg1=Prod(cc1,Set(cc)),\n cc1=P rod(Z,Union(Prod(An,tree1),tree2),\n Set(tree,card=m-2), \n Sequence(Prod(Z,Set(tree,card=m-1)))),\n tree1=P rod(Z,Union(Prod(tree1,Set(tree,card=m-1)),\n \+ Púrod(De,Set(tree,card=m)),De)),\n tree2=Prod(Z,Union(tree2,Prod(A n,tree1)),\n Set(tree,card=m-1)),\n An=Epsilon,De =Epsilon\} union Gm:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "And the g rammar for the distances to the cycles plus the length of the cycle:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 300 "Gm3:=\{fg2=Prod(cc2,Set( cc)),\n cc2=Prod(Z,Union(Prod(tree1,Set(tree,card=m-2)),\n \+ Prod(De,Set(tree,card=m-1))),\n Sequence (Prod(Z,Set(tree,card=m-1))),\n Prod(Z,An,Set(tree,card= m-1)),\n Sequence(Prod(Z,Set(tree,card=m-1))))\}\nunion \+ Gm2:" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m=2" " /%\"mG\"\"#" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "We first check on t he case " }{XPPEDIT 18 0 "m=2" "/%\"mG\"\"#" }{TEXT -1 56 " that we re cover the generating functions we had before." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "sol:=gfsolve(subs(m=2,Gm3),labelled,z);" }} {PARA 12 "" 1 "" {XPPMATH 20û "6#>%$solG<./-%#AnG6#%\"zG\"\"\"/-%\"ZGF) F*/-%#DeGF)F+/-%#ccGF)-%#lnG6#*$,&*$F*\"\"#!\"#F+F+#!\"\"F;/-%%treeGF) ,$*&F*F>,&F;F+*$F9#F+F;FF9F=/-%&tree2GF),$*&,&FCF=F*F+F+,&F>F+F:F;F>F;/-%$fg2GF)** F*F;FDF+,(FEF+F:F<*&F*F;FDF+F+F>F9F=/-%&tree1GF),$*(F*F>FDF+F9F=FF/-%$ cc2GF)*(F*F;FDF+FenF>/-%$cc1GF)*(F*F+FKF+FLF>/-%#fgGF)F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "normal(subs(sol,fg(z))-F);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "normal(subs(sol,fg1(z))-Fpl);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "nor mal(subs(sol,fg2(z))-Fplcycle);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"!" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m=6" " /%\"mG\"\"'" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The \+ grammar becomes" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "G6:=subs (m=6,Gm3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#G6G<-/%#fgG-%$SetüG6#% #ccG/%$fg1G-%%ProdG6$%$cc1GF(/%#AnG%(EpsilonG/%#DeGF4/%$fg2G-F/6$%$cc2 GF(/%%treeG-%&UnionG6$%\"ZG-F/6$FA-F)6$F=/%%cardG\"\"'/F1-F/6&FA-F?6$- F/6$F3%&tree1G%&tree2G-F)6$F=/FG\"\"%-%)SequenceG6#-F/6$FA-F)6$F=/FG\" \"&/FP-F/6$FA-F?6%-F/6$FPFen-F/6$F6FDF6/FQ-F/6%FA-F?6$FQFNFen/F;-F/6'F A-F?6$-F/6$FPFR-F/6$F6FenFV-F/6%FAF3FenFV/F+-%&CycleGFX" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "From there we compute the generating func tions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sol:=gfsolve(G6,l abelled,z):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Here is the genera ting function for the functional graphs:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "F:=subs(sol,fg(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"FG,$*$,&!$?\"\"\"\"*&%\"zGF)-%'RootOfG6#,(%#_ZG!$?(F+\"$?(*&F+F) F0\"\"'F)\"\"&F)!\"\"F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 232 "Again , the coefficients admit a closed-form which can be obtained using gfu n (it could also be derived by Lagrange's inversion formula). We rathe r compute directly the aýsymptotic expansion of the number of functiona l graphs of size " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 269 " whose nodes have an in-degree of 0 or 6 by singularity analysis. Unfortunat ely, Maple's series command is not yet able to deal with algebraic fun ctions like the RootOf above. We therefore start from the equation in \+ the system which is at the origin of the singularity:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "T:=subs(sol,tree(z));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"TG-%'RootOfG6#,(%#_ZG!$?(%\"zG\"$?(*&F+\"\" \"F)\"\"'F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "tree(z)-subs (gfeqns(G6,labelled,z),tree(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,( -%%treeG6#%\"zG\"\"\"-%\"ZGF&!\"\"*&F)F(F$\"\"'#F+\"$?(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eq:=subs(tree(z)=y,Z(z)=z,\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG,(%\"yG\"\"\"%\"zG!\"\"*&F(F'F& \"\"'#F)\"$?(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "diff(eq,y) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*&%\"zGF$%\"þyG\"\"&#!\" \"\"$?\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve(\{\"\",\" \},\{y,z\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<$/%\"zG,$-%'RootOfG6# ,&*$%#_ZG\"\"$\"\"\"!#7F.#\"\"&\"\"'/%\"yGF'<$/F4-F(6#,&F+F.\"#7F./F%, $F7F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "It follows that the domi nant singularity is at " }{XPPEDIT 18 0 "z=5/6*(12)^(1/3)" "/%\"zG*(\" \"&\"\"\"\"\"'!\"\")\"#7*&\"\"\"F&\"\"$F(F&" }{TEXT -1 8 ", where " } {XPPEDIT 18 0 "y=12^(1/3)" "/%\"yG)\"#7*&\"\"\"\"\"\"\"\"$!\"\"" } {TEXT -1 29 ". The singular expansion of " }{XPPEDIT 18 0 "y(z)" "-% \"yG6#%\"zG" }{TEXT -1 65 " can be computed at that point by power ser ies reversion (we set " }{XPPEDIT 18 0 "theta=sqrt(1-z/rho)" "/%&theta G-%%sqrtG6#,&\"\"\"\"\"\"*&%\"zGF)%$rhoG!\"\"F-" }{TEXT -1 7 " where \+ " }{XPPEDIT 18 0 "rho" "I$rhoG6\"" }{TEXT -1 49 " is the singularity, \+ which helps series a little)" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "sing:=5/6*12^(1/3):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 47 "ÿseries(subs(z=sing*(1-theta^2),eq),y=12^(1/3));" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#+1,&%\"yG\"\"\"*$\"#7#F&\"\"$!\"\",&F' F&*&F(F),&F&F&*$%&thetaG\"\"#F+F&F+\"\"!F/\"\"\",$*&F(#F1F*F.F&#!\"&\" #C\"\"#,$F-#F8\"#=\"\"$,&F7F&F/#\"\"&F9\"\"%,$F5#F+\"$W\"\"\"&-%\"OG6# F&\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "\{solve(\",y)\}; " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<$+/%&thetaG*$\"#7#\"\"\"\"\"$\"\" !,$*&\"#5#F)\"\"#F'F(#!\"\"\"\"&\"\"\",$F&#F2\"#:\"\"#,$F-#!#r\"$+*\" \"$,$F&#\"\"(\"%vL\"\"%-%\"OG6#F)\"\"&+/F%F&F+,$F-#F)F3\"\"\"F5\"\"#,$ F-#\"#rF<\"\"$F>\"\"%FC\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "sery:=op(select(proc(s,u) evalb(signum(coeff(s,u,1))=-1) end,\", theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%seryG+/%&thetaG*$\"#7# \"\"\"\"\"$\"\"!,$*&\"#5#F*\"\"#F(F)#!\"\"\"\"&\"\"\",$F'#F3\"#:\"\"#, $F.#!#r\"$+*\"\"$,$F'#\"\"(\"%vL\"\"%-%\"OG6#F*\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "series(subs(T=sery,z=sing*(1-theta^2),F), theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+-%&thetaG,$*$\"#5#\"\"\"\" \"##F)F'!\"\"#\"\"%\"#:\"\"!,$F&#\"#Z\"$+'\"\"\"#\"$;'\"%vL\"\"#-%\"OG 6#F)\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Hence the number of \+ functional graphs with in-degree in \{0,6\} is asymptotically:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "as_fg:=coeff(\",theta,-1)/sq rt(Pi)*n^(-1/2)*sing^(-n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&as_fg G,$**\"#5#\"\"\"\"\"#%#PiG#!\"\"F*%\"nGF,),$*$\"#7#F)\"\"$#\"\"&\"\"', $F.F-F)#F)F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "We next consider \+ the distance to the cycle" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Fpl:=subs(sol,fg1(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$FplG, $*,%\"zG\"\"\"-%'RootOfG6#,(%#_ZG!$?(F'\"$?(*&F'F(F-\"\"'F(\"\"%,&F)! \"&F'F1F(,,!#IF(*&F'F(F)\"\"&!#E*&F'\"\"#F)F2\"#***&F'\"\"$F)F>!$E\"*& F'F2F)F;\"#a!\"\",&!$?\"F(F7F(FB\"$]\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "series(subs(T=sery,z=sing*(1-theta^2),Fpl),theta);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#+'%&thetaG#\"\"\"\"#?!\"%-%\"OG6#F&! \"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "And the length of the cycl e:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Fpl2:=subs(sol,fg2(z) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fpl2G,$*,%\"zG\"\"#-%'RootOfG 6#,(%#_ZG!$?(F'\"$?(*&F'\"\"\"F-\"\"'F1\"\"$,&F)!\"\"F'F1F1,*!#SF1*&F' F1F)\"\"&\"\"(*&F'F(F)\"\"%!#=*&F'F3F)F3\"#7F5,&!$?\"F1F8F1F5!%S9" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "series(subs(T=sery,z=sing*(1 -theta^2),Fpl2),theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+)%&thetaG# \"\"\"\"#?!\"%,$*$\"#5#F&\"\"##!#6\"$+$!\"$-%\"OG6#F&!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Thus the distance to the cycle and the le ngth of this cycle are both asymptotic to:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 48 "asympt(coeff(\",theta,-4)*n*sing^(-n)/as_fg/n,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"#5#\"\"\"\"\"#%#PiGF&*$%\"nG! \"\"#F,F(#F'\"#?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 152 "Thus the rat io between this variant of Pollard's algorithm and the original one is in both cases (number of steps to the cycle and length of the cycle): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "ratio:=simplify(map(asy mpt,[as_cost_1/\",as_cost_2/\"/2],n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ratioG7$*$\"\"&#\"\"\"\"\"#F&" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Conclusion" }}{PARA 0 "" 0 "" {TEXT -1 374 "This workshee t shows that some algorithms whose complexity analysis does not reduce to counting the number of subcomponents in some kind of recursive com binatorial structure can sometimes be treated by combstruct using seve ral marks. This is in particular true of many recursive searching algo rithms whose complexity is related to the path length of an underlying structure." }}}}{MARK "1 0 0" 29 }{VIEWOPTS 1 1 0 1 1 1803 } \",theta,-4)*n*sing^(-n)/as_fg/n,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"#5#\"\"\"\"\"#%#PiGF&*$%\"nG! \"\"#F,F(#F'\"#?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 152 "Thus the rat io between this variant of Pollard's algorithm and the original one is in both cases (number of steps to tface J@Ü_facilitP '‡/ ^!vkw facilitat:yfaclÕ\facpÕ\facqÕ\fact+à"ü;†P¼QÕ\Îjòs:yfacto"factorgR¿ £ èÍ "µ%3(‡/7½8J@àA .F†P‚Q¼QÎjBkëlÊo!vkwîx:yfactorgàA factorgfqfrfÎjfactoriaÕ\ factorial ¿  ,†PÕ\dÎj factorialg'#à¿ £ Í J@¼QÕ\Îj :y factorizat ˆ;jfadÎjfadlfjÕ\fadmÕ\fadpfÕ\fadsÕ\fadvÕ\faeÕ\fael ÎjëlfaepfÕ\fafTŠ ¿ £ 9¯µ%ú+½4W5778½8ˆ;$@J@ àA.F†P¼Q‰\ ^Ü_dÎjBkëlypt:yfafa†PÎj:yfafaf Îj:yfafap9fafbfÜ_fafc:yfafcf$@fafd:y ^•decalphaunixmaplinputcourimathtimehyperlinkcommoutputhelpheadnormallistitembulletashintroductgroebnpackagbaseusagcallsequencgroebnerfunctargssynopsidescriptcollectroutindoingbasiscalculatskewalgebralikeweylorecorrespondmoduldgcanalsousedcaseusualcommutatpolynomialuseeithdefinaloncommandwithgroebnallfunctionwithacceswithoutusingcommandformavailablarfollowtermordtestordgroebnerleadcoeffgroebnerleadmonerleadtermeadtermnormalfnormalfreducinterreducegbasipretendpretndgsolvgsolvespolunivpofglmalgofinitsolvablhilbertdimhilbertdihilbertpohilbertpolyhilbertserhilbertserthesmakebasirelatfacilitavailablpolynomialsealgebradetailsuchionsalgebraicmodularcoefficienttermorderdeclartermorderreturntabldescribneededotheralternativeshortordsimpltypeshorttermordcheckgivenmanypreorderingincludtotaldegrelexicographiceliminatmatrixleadcoeffleadmleadtermcomp3siioëlsikfraëlsil"silon"W5Îjëlòssilong{q8ëlsim:"†P_similarOÄ3àŠ £ Ƶ%3(‡/†P Så]Ü_Îjëloòs:ysimp7 ¿ R6ˆ;Â;ü;àA»M*O¼QjBkÊosimpl_-à¿ £ µ%'‡/W5ˆ;J@˜@†P¼QÕ\ ^Ü_ÎjBkëlòs!vkw:ysimplest:ysimplidsimplif/$à¿ ‚ "J@àA†PÎjBkòs:ysimplifi ¼QÌ^ÎjÊo!v:y simplifygdÎj:y»Iproduc/Š ç £ ½:k¿"Ë-ƒ4W56˜@»MãM*OªYà^Ü_`§bBkëlÔmÊoypŸp!vîx:yproduct¿ß3.¿ ‚ 9Æ:KÍ " µ%'3(`+ƒ4½4R67½8S:ˆ; ü; <¥=˜@àA.F»I»M *O ‚Q¼QªY‰\Õ\ ^à^Ü_*d†ioÊo9qòs!vëlW5¼QÕ\Îjîxgnuëlgo7#à¿ ‡/W5ÀD¼QÕ\ÎjëlÊotîx:ygoalŠ ¿ Õ\jÎj goemetricogoes# Š ¿ ïJ@LDÕ\òsgojŠ gommr"gomoÕ\gonkÕ\òs!vgooogood àŠ £ †P¼Q:ygop¿ goqtëlgordonµ%gorithm ‡/BkgosaŠ got¿ gotten¿ gourdon Š †Pgouyou•rgovcëlgozÕ\gozkrëlgp+¿ "ú+W5Õ\jÎjëlîxgpaÎjgpf"gpjyÕ\gplfesf"gppÕ\gppnÕ\gpro ¿ ëlgprobëlgprobgëlgpv"gq¿ W5Õ\ëlgqbfw¿ gqi¿ gqlaw¼QgqqÕ\gqqmgëlgqqyëlgqqyyëlgqt ël background•rbackwardÜ_†i9qbad †PÁqbafÕ\bal‡/balanc ˜@Îjball ¼Q!vbanana'bananag'baq¿ barbour:ybarkatou•rbasP ˆ;Êo!vbasek/¿ P ŒÆ?èkÍ 3(‡/½4R6½8ˆ;J@¶J»MãM ^Ü_ÎjëlÔm}o•rkwbasedW$࣠3(Ô+‡/R6½8àA.F†P!Q‰\ ^†ijÎjBkÊoòs!v:ybasegf Ö:ybasiSF¿ P Œì?èÍ 3(ú+‡/ ½4Þ5½8<J@CN ^ Áqkw :ybasic3Æ'†P¼QÜ_jÎjBkòs!v:ybasicalbatŠ batxÕ\bau¿ bazkÕ\bb iŠ £ ½Õ\bbbbÕ\ bbbbbbbbbb¿ atoric Ü_:yatreÕ\atrix¶Jatrixg‡/att ç ŒkãMÜ_Ômattach Õ\òs:yattack ¿ ëlattain¿ ç òs attainabl‰\attemptà$@†Pòsuyattent Š :yatternëlattestÎjattrib $@‰\attribu‰\attribut+5ç 9Æ'$@ A‰\Ü_•r!vauC3à¿ ‚ "Ë-q8.F¼QÕ\˜]ël•r!vaubhÕ\audzhÕ\aufÕ\aug Õ\ëlaugustÕ\auldëlaupbëlautŠ Ë-àAÀDauthor#£ P µ%àAjo•r!vauto3)½{Ë-/78q8ÀD˜]tŠ} autocarprod 3½Š}autoco3autocom½autocomb P !vlindicat'¿ :J@-BÕ\BkëlòsypŸp9qÁq sòs!vkwuy€ tuned¼QtungstenŠ tunrmsgÕ\tunskvëltuoplÕ\tupl‡/ture'¼Qël•rturhëlturn7Š ¿ 9Í <$@‚QÕ\Bkëlo!v:ytutëltutionÕ\tutorial!vtvàŠ Õ\ëltvcÕ\tvjëltvpëltvsÕ\tvvÕ\tw ¿ ü;J@˜@Õ\Îjtwic"Õ\ëltwjfëltwndnëltx¿ :V1W5.F Õ\ëltxg.Ftxgf.FtxiëltxsiÎjtxt¿ strategàstrehlàA stributional†PstrictÎjstrikstring7Š 'ˆ;¶J*Oý\§b†iÎjëlŸpIv:ystripŠ strong¿ àA!vstruý\struc'¼Q•rstruct3àç W5 A¼Qý\§bBkŸp!vIvfpfcëlfpfcfqfcfpfdëlfpfdfdfdfrfhflcëlfpfenBkfpfgŠ fpfhnfJ@ fpfhoffof½8fpfjf ¿ J@fpfrBkfpfxfràAfphqw¿ fpl Bk fplcyclBk fplcyclegBkfplgBkfpm¿ "ÎjfpmaàAfpmtf¿ fpq¿ fpxfduÕ\fpynjf"fq79Š ¿ "‡/ˆ;J@àA.FÕ\ Bkël òs:yfqaÕ\fqaf¿ fqbfëlëlpqgW5pqgoezk¿ pqh¿ pqlfhv"pqmnÕ\haven†PhavingŠ  ^ÎjëlhavtëlhawjfduÕ\hawkÕ\haye¼Qhazëlhb ¿ "àAÕ\ÎjëlhbfnnaÕ\hbgëlhbheëlhbjëlhblÕ\hblgÎjhblv¿ hbmëlhbmf¿ hbnÕ\hbnsbàAhboël hbourhoud¿ hbpW5hbqëlhbqgëlhbrÕ\hbreëlhbtcfd¿ hbwëlhbyqwcëlhbzmfhv"hbzw¿ hcà¿ ".FÕ\ëlhcf".FÕ\ëlhcffpëlhch†Phci Õ\hcjÕ\hckw à¼QhclëlhclmëlhcnÕ\ hcnqvmpxfëlCN&decalphaunixmaplinputcourimathtimehyperlinkcommoutputourihelpheadnormalbulletitemfunctgroebnsolvabldecidgivenalgebraicsystemalgebraicalconsistusagcallsequencfgxgparametersetlistpolynomialindeterminatsynopsidescriptcommandsolvablegidesusingtotaldegrebasipolynomialwithrespectalgebraicalateastsolutpermutatariablusedassociatcomputatspecifiindetindetsgassumedpartroebnpackagcanformonlyfterperformalwayaccesslongolvablexamplnfnistruegopfalsseealsoalsofinitunivpogsolvssing_1,autocombnoncrossing_2,autocombÕ\permutations,autocomb:ypol_to_sys,Mgfunpoly_algebra,Ore_algebra¥=polymer_tilings,autocombŠ pretend_gbasis,Groebnerkwqshift_algebra,Ore_algebraR6interpretsequencinitialvalucomputthemproceedistepotherproceedlengthstartnodeeventualreachboundintersectgorithmstopunderaveragrequiredrelatparameterdistanccycleiiclespathplanarbinarquestabovdeterminatveragconcentratimilarbutproblemcomputatfromrootclassicalparametinternallengthsumromnbinardescribbinepsiloncountgivenconstitutedwellknowncatalannumbersequnlabellsizeyehuswntakeproducusingreturnsipltypenameformelseformopnsizevalsubsendbijectionbranchleafoncebelowgeneratdrawsizetgprodgepsilongshalltotaldividthexperimentalfixedsmallallstructallstructuremapizeiplgactualmoreefficienttogethmanyresultabilitdealmarkatomdeaconsistwritsuchnumberobjectexacttreescombinatorialextendleadbasednthemodifiaccountdecoratputtancestordedescendantbelongsamewayscoutcorrespondsummleftrightleftrightbintrepsiloncumulatnowlessseconypyhzgangdegffefeffwfcfgofgnffufapffunctionaryviacombstructgfsolvcumulatserisgqwnnzwqwzjswklqemsocohfimctsyblzujogfplfplgsplplsplgenkgrpoxvyxiwidmqkwbgagyarxnsalrcqytydpvlzejotsimpratiocoefficientcoeffjcrhqaddvlywrafqlfjovzbncycgvkncgxjjowckoimsfhcflyihxqmrnszorohevalfdgjqvjflvfqklpdfqvkfzsfyvlqvokcgfdfmidrdbpiempiricalesegrowslightfastlineartreeclosestablishrigorouslgeneratalgebraicsatisfdifferentialquationholexprtodiffholexprtodiffeqdifferialequationrecurrencsatisfitaylorcoefficiobtaindiffeqtorecdiffeqtorecturnfallclasrsolvsolutequationsdeqfeqdeqfgdiffgdeqfpldeqfplgzgfeffdfdfdcffcfdrecfrecfgugrecfplrecfplggforderobviouwillablesolvcatcatggammagpighowevsurprissolutiorddueimplementatrecentpetkovsekhypergeometricsolutionplgaftersomecleaniprovpropositaverauniformbinomialbinomialgasymptoticbehaviourwithinasymptcommandexpcomebackfungraphgsetgcomponentgcyclegeeguniongcardgprocesisolatbetweeninstancnonnpbinnpbingbintreegindicatrespectpositsubtredoeslabellcidealwayleftmostinsteaconsidercasintreagaincheckcoinlocalubsindetallstructslrocesreadiinvolvdecompositsetsmarkedpartfungrapntconnectednunionocedurtheirouiplfgiplccniplcccciplbtniplbtbtipltnipltsubtestprintffionalquitodgonlyreasonablamountphsortallstrucconverthandountminutalsogetfunctionacomputeapplyingdifferentialequatdifffafrfunfortunateweaknescurrfoundrsolvegfpffhnfhelptakingparitdeqfpfbnbfggtotsimplifrecfplsummarsqrtgammasqrtgngfexpanspigfcostourprobabilisticstaghoasymptoticalcgpgfbotbeformightcorpriorisufficiobeyinsteadmodifancestelemovernewremovtypidenticalanythcomponentfplcyclfplcyclegconclustheorembetweenwhosfactorizatnasymptoticalbehavlikeverificatsecondcostptjrekmarithmeticmodelextensmgprioriadvancconjecturhisreducolynomialbrentideafermatknowconsidgmrodrammarpluscoverhadsolsolgccgflngtreegffcffoffdffenffkfflffggfnormalbecomccgtreegsequencegfpfpfenfqfqfnfenfpfrfenfvfafcyclegfxfunctionabellgeneratingrootofgadmitgfulagranginversformularathsingularitunfortunatelyyetfunrootofsystemorigingfeqneqgygfdominantsingularpowerseriesreversthetalittlsingthetagrfseryevalbsignumseryghencsqrtnextsfratiooriginalasymptratiogworkshesubcomponentkindrecursbinatorialsometimtreatseveralparticulartruesearchalgorithmunderlyparitdeqfpfbnbfggtotsimplifrecfplsummarsqrtgammasqrtgngfexpanspigfcostourprobabilisticstaghoasymptoticalcgpgfbotbeformightcorpriorisufficiobeyinsteadmodifancestelemovernewremovtypidenticalanythcomponentfplcyclfplcyclegconclustheorembetweenwhosfactorizatnasymptoticalbehavlikeverificatsecondcostptjrekmarithmeticmodelextensmgprioriadvancconjecturhisreducolynomialbrentiøŠ{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 2 0 0 0 0 0 0 0 } {CSTYLE "" -1 259 "" 1 12 0 0 0 0 1 2 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 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 16 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 2 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 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 1 12 0 0 0 0 1 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 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 }{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 "T ext Output" -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 0 -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 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 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 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 "" 1 18 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 }{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 {EXCHG {PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 256 36 "ENUMERA TION OF PLANAR CONFIGURATIONS" }}{PARA 260 "" 0 "" {TEXT 263 25 "IN CO MPUTATIONAL GEOMETRY" }}{PARA 263 "" 0 "" {TEXT -1 0 "" }}{PARA 261 " " 0 "" {TEXT 265 17 "Philippe Flajolet" }}{PARA 262 "" 0 "" {TEXT -1 29 "(Version of January 15, 1997)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 9 "Consider " }{XPPEDIT 18 0 "n" "I\"nG6\"" } {TEXT -1 256 " points on a circle that define a convex polygon. The en umeration of geometric configurations that can be superimposed on thes e points has a dignified history. In 1753, Euler solved the problem of counting the number of possible triangulations of a convex " } {XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 178 "-gon while inventing the c oncept of generating function for this particular combinatorial enumer ation problem. Since then, many configurations have been enumerated; s ee Comtet's " }{TEXT 257 24 "Advanced Combinatorics, " }{TEXT -1 71 "p . 74 for a discussion of works of Prouhet 1886, Jordan 1920, Guy 1967. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 356 "Till recently, any new enumerative result was basically a research paper, \+ and some of the applications discussed below have required several pag es of recurrence manipulations. In this report, we take inspiration fr om a recent memo of Noy (September 1996) and show that many such resul ts can be derived automatically using the Maple system and the package s " }{HYPERLNK 17 "Combstruct" 2 "combstruct" "" }{TEXT -1 5 " and " } {HYPERLNK 17 "Gfun" 2 "gfun" "" }{TEXT -1 53 ". This worsheet is meant as a simple introduction to " }{HYPERLNK 17 "Combstruct specification s" 2 "combstruct[specification]" "" }{TEXT -1 71 " as well as to basic experimental computations that may accompany them." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 34 "Euler's counting of triangulations" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Problem" }}{PARA 256 "" 0 "" {TEXT -1 47 " Find the number of ways of cutting up a convex " }{XPPEDIT 18 0 "``(n+ 2)" "-%!G6#,&%\"nG\"\"\"\"\"#F'" }{TEXT -1 34 "-gon into n triangles b y means of " }{XPPEDIT 18 0 "``(n-1" "-%!G6#,&%\"nG\"\"\"\"\"\"!\"\"" }{TEXT -1 28 " non-intersecting diagonals." }}{PARA 258 "" 0 "" {TEXT 262 73 "Here are examples of triangulations of the octogon and of the icosagon.\n" }{INLPLOT "66-%'CURVESG6#7$7$$\"3\\+++D[0rq!#=$\"3>+++*z !3rqF*7$$!3\")*****H50Kn$!#B$\"\"\"\"\"!-F$6#7$F-7$$!3#)*****>x162(F*$ \"3v*****>&)G52(F*-F$6#7$F7F'-F$6#7$F77$$!\"\"F3$!3&******p?5kM(F0-F$6 #7$FB7$$!30+++yG+rqF*$!3W*****\\uK62(F*-F$6#7$FJF7-F$6#7$7$$\"3=+++=(e 62(F*$!3M+++/p(42(F*7$$\"2*******)*********!#<$\"3.+++T?Gp9!#A-F$6#7$F Z7$$\"3_******H4&42(F*$\"3))*****>p%=rqF*-F$6#7$F^oFU-F$6#7$7$$\"3$*** ***4`h>5\"Fjn$!2*******)*********FgnFU-F$6#7$FUF^o-F$6#7$F^oFio-F$6#7$ FJFio-F$6#7$FioF^o-F$6#7$F^oFJ-F$6#7$7$F1F3F'-F$6#7$F'FU-F$6#7$FUF`q-% *AXESSTYLEG6#%%NONEG-%'COLOURG6&%$RGBGF3F3$\"*++++\"!\")" 2 217 215 215 2 0 1 0 2 6 0 1 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20040 0 12020 0 0 0 255 0 0 255 1 0 0 0 0 91 100 0 0 0 0 0 0 }{INLPLOT "6J-%'CURVESG6#7$7$$\"\"\"\"\"!F*7$$\"3o*****H*Gc5&*!#=$ \"3!******4$p9!4 $F.$\"3i*****RCu0^*F.-F$6#7$FDF4-F$6#7$FD7$$!3\")*****H50Kn$!#BF(-F$6# 7$FO7$$!3*)*****\\'=@!4$F.$\"3v*****>a^0^*F.-F$6#7$FVFD-F$6#7$FV7$$!3h +++ET*y(eF.$\"3,+++<(R,4)F.-F$6#7$F[o7$$!3A+++*[/-4)F.$\"3^*****f(\\!y (eF.-F$6#7$Fco7$$!3f+++%f&e5&*F.$\"3&)*****>12,4$F.-F$6#7$F[pF[o-F$6#7 $FVF[p-F$6#7$FDF[p-F$6#7$F4F[p-F$6#7$F[p7$$!\"\"F*$!3&******p?5kM(FR-F $6#7$F4F_q-F$6#7$F_q7$$!3\")*****4>S0^*F.$!3))******)zY-4$F.-F$6#7$Fjq 7$$!3\"******f7=,4)F.$!3E+++VQ#z(eF.-F$6#7$Fbr7$$!3')******e_xxeF.$!3C +++zgA!4)F.-F$6#7$FjrFjq-F$6#7$F_qFjr-F$6#7$Fjr7$$!3')*****z7s+4$F.$!3 _+++Xpf5&*F.-F$6#7$Fhs7$$\"3$******4`h>5\"!#A$!2*******)*********!#<-F $6#7$F`tFjr-F$6#7$F_qF`t-F$6#7$F`t7$$\"3')*****Ht\"G!4$F.$!3()******R) G0^*F.-F$6#7$F_qF`u-F$6#7$F`u7$$\"3\"*******fN&z(eF.$!3\")*****\\`'4!4 )F.-F$6#7$F_qF[v-F$6#7$F[v7$$\"3D+++pwC!4)F.$!3@+++UbuxeF.-F$6#7$F_qFf v-F$6#7$F4Ffv-F$6#7$F+Ffv-F$6#7$F'Ffv-F$6#7$Ffv7$$\"3Q+++&H31^*F.$!3z* ****H>P+4$F.-F$6#7$F'Fjw-%*AXESSTYLEG6#%%NONEG-%'COLOURG6&%$RGBGF*F*$ \"*++++\"!\")" 2 218 217 217 2 0 1 0 2 6 0 1 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20040 0 12020 0 0 0 255 0 0 255 1 0 0 0 0 98 80 0 0 0 0 0 0 }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Specification" }}{PARA 0 "" 0 "" {TEXT -1 49 "A triangulation deco mposes into three components:" }}{PARA 0 "" 0 "" {TEXT -1 115 " - the \"root\" triangle uniquely defined uniquely by the fact that it c ontains the edge with smallest numbers (" }{XPPEDIT 18 0 "0,1" "6$\"\" !\"\"\"" }{TEXT -1 12 " initially);" }}{PARA 0 "" 0 "" {TEXT -1 105 " \+ - the left and right subtriangulations defined by their position \+ with respect to the root triangle." }}{PARA 0 "" 0 "" {TEXT -1 19 "Fir st, we load the " }{HYPERLNK 17 "combstruct" 2 "combstruct" "" }{TEXT -1 9 " package:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "with(comb struct);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.%+allstructsG%&countG%%d rawG%)finishedG%'gfeqnsG%)gfseriesG%(gfsolveG%,iterstructsG%+nextstruc tG%,prog_gfeqnsG%.prog_gfseriesG%-prog_gfsolveG" }}}{PARA 0 "" 0 "" {TEXT -1 54 "In writing a specification, we take the atomic symbol " } {XPPEDIT 18 0 "Z" "I\"ZG6\"" }{TEXT -1 23 " to denote a traingle, " } {XPPEDIT 18 0 "T" "I\"TG6\"" }{TEXT -1 179 " being an arbitrary triang ulation. One must be careful to isolate one-sided triangulations, wher e either the left or the right subtriangulations may be empty. Thus, t he grammar is" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "triang:=[T, \{T=Union(Z,Prod(Epsilon,Z,T),Prod(T,Z,Epsilon),Prod(T,Z,T))\},unlabel led]:" }}}{PARA 0 "" 0 "" {TEXT -1 65 "This specification is clearly r eminiscent of binary search trees." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Counting. " }}{PARA 0 "" 0 "" {TEXT -1 59 "Now, we can count th e number of triangulations of size 100:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "count(triang,size=100);" }}{PARA 0 "" 0 "" {TEXT -1 120 "Here the system takes a couple of seconds to set up (once and for all!) complete counting tables till size equal to 100." }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"Z?$4fuQ:_P3UK15u+2qro'\\J,4Z*>l*)" }}}{PARA 0 "" 0 "" {TEXT -1 39 "In particular, the first few values are" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "seq(count(triang,size=i),i=0 ..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "67\"\"!\"\"\"\"\"#\"\"&\"#9\"# U\"$K\"\"$H%\"%I9\"%i[\"&'z;\"&'ye\"'7!3#\"'+Hu\"(SWn#\"(X[p*\")qwNN\" *!zW'H\"\"*+(QwZ\"+!>jsw\"\"+?/7kl" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 151 "These numbers are known as the Catalan numbers in the honor of the French and Belgian mathematician who worked out their main proper ties in the 1850's." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Random generation." }}{PARA 0 "" 0 "" {TEXT -1 349 "We can draw a triangulation at random amongst those of a ny given size. Note that uniformity is granted a priori by the algorit hms contained in the combstruct package. Naturally, the output is in t he format dictated by the specification. Here is a triangulation of si ze 10, that is to say, comprised of 10 triangles, and corresponding to a dodecagon." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "draw(triang ,size=10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%ProdG6%-F$6%%\"ZGF(-F $6%-F$6%%(EpsilonGF(-F$6%F-F(F(F(F-F(-F$6%F.F(F-" }}}{PARA 0 "" 0 "" {TEXT -1 210 "Since the objects generated are standard Maple objects, \+ we can manipulate them and in particular construct plots to visualise \+ them. Here is a set of procedures to construct the list of edges of a \+ triangulation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "size:=proc (t) convert(map(size,t),`+`) end: size(Epsilon):=0: size(Z):=1:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 390 "set_edges:=proc(tree,org,ed ge_set) local se1,se2;\n if tree=Epsilon then\n org,org,edge _set union \{[org,org+1]\}\n elif tree=Z then\n org,org+1,ed ge_set union \{[org,org+1],[org+1,org+2],[org+2,org]\}\n else\n \+ se1:=set_edges(op(1,tree),org,edge_set);\n se2:=set_edges(o p(3,tree),se1[2]+1,se1[3]);\n se1[1],se2[2],se2[3] union \{[se1 [1],se2[2]+1]\}\n fi\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "plott:=proc(n) plot([op(map2(map,[cos,sin],expand(map(`*`,set_e dges(draw(triang,size=n-2),0,\{\})[3],2*Pi/n))))],color=blue,axes=NONE ) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "plott(20);" }} {PARA 13 "" 1 "" {INLPLOT "6I-%'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`P:&H;l0^*F*7 $$!\"\"\"\"!F;-F$6#7$F8F'-F$6#7$F57$F($\"3^u%\\P%*p,4$F*-F$6#7$FBF8-F$ 6#7$7$F.$\"3PJZ#H__y(eF*F5-F$6#7$F57$F0$\"3^u%\\P%*p,4)F*-F$6#7$FQFK-F $6#7$F57$FCF6-F$6#7$7$FLFRF5-F$6#7$7$FRFLF5-F$6#7$FhnFZ-F$6#7$FZ7$F;$ \"\"\"F;-F$6#7$FcoF5-F$6#7$F\\oFhn-F$6#7$F-F8-F$6#7$F5F--F$6#7$F\\o7$F 0F.-F$6#7$7$FdoF;7$F6F+-F$6#7$Fip7$F6FC-F$6#7$F^qF\\o-F$6#7$Fip7$FRF0- F$6#7$FeqFjp-F$6#7$F^qFeq-F$6#7$7$F;F9Feq-F$6#7$F\\oFeq-F$6#7$7$FLF.F_ r-F$6#7$FfrFeq-F$6#7$F_r7$FCF(-F$6#7$F]sFfr-F$6#7$7$F+F(F_r-F$6#7$F\\o F_r-F$6#7$F5Fep-F$6#7$FepFds-F$6#7$F-Fep-F$6#7$F\\oFds-F$6#7$FKFB-%'CO LOURG6&%$RGBGF;F;$\"*++++\"!\")-%*AXESSTYLEG6#%%NONEG" 2 588 588 588 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 16360 255 0 0 0 0 0 0 }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 22 "Exhaustive generation." }}{PARA 0 "" 0 "" {TEXT -1 34 "With the December 1996 release of " }{HYPERLNK 17 "Combstruct" 2 "combstruct" "" }{TEXT -1 219 ", we may also list exhautively all co nfigurations of a given size. This proves useful for exhaustive testin g of small cases, though there are physical limitations owing to the \+ exponential growth of the number of cases." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "tr4:=allstructs(triang,size=4);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%$tr4G70-%%ProdG6%%(EpsilonG%\"ZG-F'6%F)F*-F'6%F)F*F *-F'6%-F'6%-F'6%F*F*F)F*F)F*F)-F'6%-F'6%F)F*F3F*F)-F'6%F-F*F*-F'6%F*F* F3-F'6%F3F*F*-F'6%F)F*-F'6%F*F*F*-F'6%F)F*-F'6%F-F*F)-F'6%F+F*F)-F'6%F *F*F--F'6%FAF*F)-F'6%F)F*F1-F'6%F)F*F7-F'6%FEF*F)" }}}{PARA 0 "" 0 "" {TEXT -1 34 "We found 14 objects, as we should." }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 30 "Generating function equations." }}{PARA 0 "" 0 "" {TEXT -1 152 "Again, this uses the December 1996 release of Combstruct . We may form the generating function equations associated to a given \+ specification, as follows:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "gfeqns(op(2,triang),unlabelled,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/-%\"ZG6#%\"zGF(/-%\"TGF',(F%\"\"\"*&F%F-F*F-\"\"#*&F*F/F%F-F- " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "gfsys:=gfsolve(op(2,tri ang),unlabelled,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&gfsysG<$/-% \"TG6#%\"zG,$*&F*!\"\",(\"\"\"F/F*!\"#*$,&F/F/F*!\"%#F/\"\"#F-F/F4/-% \"ZGF)F*" }}}{PARA 0 "" 0 "" {TEXT -1 119 "Here, the solution could be obtained explicitly. Correctness can be checked by a Taylor expansion , comparing with what " }{HYPERLNK 17 "combstruct[count]" 2 "combstruc t[count]" "" }{TEXT -1 10 " gives us." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "op([1,2],\"); series(\",z=0,12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"zG!\"\",(\"\"\"F(F%!\"#*$,&F(F(F%!\"%#F(\"\"#F&F (F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+9%\"zG\"\"\"\"\"\"\"\"#\"\"#\" \"&\"\"$\"#9\"\"%\"#U\"\"&\"$K\"\"\"'\"$H%\"\"(\"%I9\"\")\"%i[\"\"*\"& 'z;\"#5-%\"OG6#F%\"#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "se q(count(triang,size=n),n=0..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-\" \"!\"\"\"\"\"#\"\"&\"#9\"#U\"$K\"\"$H%\"%I9\"%i[\"&'z;" }}}{PARA 0 "" 0 "" {TEXT -1 94 "Alternatively, an arbitrary system of generating fun ction equations can be solved by means of " }{HYPERLNK 17 "combstruct[ gfseries]" 2 "combstruct[gfseries]" "" }{TEXT -1 40 ", even when no cl osed form is available." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "g fseries(op(2..3,triang),z,[[z]],6);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #-%&TABLEG6#7$/-%\"ZG6#%\"zG+%F+\"\"\"\"\"\"/-%\"TGF*+/F+F-\"\"\"\"\"# \"\"#\"\"&\"\"$\"#9\"\"%\"#U\"\"&-%\"OG6#F-\"\"'" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 "Noy's counting of non-crossing trees" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Problem." }}{PARA 259 "" 0 "" {TEXT -1 96 "Find the number of trees (called non-crossing trees) that can be b uilt on the n vertices on the " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 51 "-gon, assuming that no edges of the tree intersect." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "Specification." }}{PARA 0 "" 0 "" {TEXT -1 68 "View node number !0 as the root, the vertices being number from 0 to " }{XPPEDIT 18 0 "n-1" ",&%\"nG\"\"\"\"\"\"!\"\"" }{TEXT -1 82 ". An arbitrary number of edges connect the root to other nodes. At each such node " }{XPPEDIT 18 0 "m" "I\"mG6\"" }{TEXT -1 93 " conn ected to the root, there is a \"butterfly\", defined by two noncrossin g trees attached to " }{XPPEDIT 18 0 "m" "I\"mG6\"" }{TEXT -1 49 " (on e non-crossing tree to the left of the edges " }{XPPEDIT 18 0 "<0,m>" "-%-anglebracketG6$\"\"!%\"mG" }{TEXT -1 34 ", the other to the right) . We let " }{XPPEDIT 18 0 "T" "I\"TG6\"" }{TEXT -1 33 " denote non-cro ssing (NC) trees, " }{XPPEDIT 18 0 "F" "I\"FG6\"" }{TEXT -1 49 " denot e forests defined by pruning the root of a " }{XPPEDIT 18 0 "T" "I\"TG 6\"" }{TEXT -1 11 "-tree, and " }{XPPEDIT 18 0 "B" "I\"BG6\"" }{TEXT -1 47 " denote a butterfly. The specification results." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "nctree:=[T,\{T=Prod(Z,F),F=Sequence (B),B=Prod(F,Z,F)\},unlabelled]:" }}}{EXCHG {PARA 0 "> " 0 "" {"MPLTEXT 1 0 34 "seq(count(nctree,size=i),i=0..20);" }}{PARA 12 "" 1 " " {XPPMATH 20 "67\"\"!\"\"\"F$\"\"$\"#7\"#b\"$t#\"%G9\"%_x\"&jK%\"'vmC \"(:2V\"\"(SYT)\")3r1]\"*s0$3I\"+?lwA=\",kcvC6\"\",f\\vG$o\"-N`aI?U\". lE/J'>E\"/+.HAHL;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "Random generation, exhaustive generation." }} {PARA 0 "" 0 "" {TEXT -1 32 "We draw objects at random with " } {HYPERLNK 17 "combstruct[draw]" 2 "combstruct[draw]" "" }{TEXT -1 27 " . Here is a random NC-tree." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "draw(nctree,size=8);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%%ProdG6$ %\"ZG-%)SequenceG6$-F$6%%(EpsilonGF&-F(6#-F$6%-F(6#-F$6%-F(6#-F$6%-F(6 #-F$6%F,F&F,F&F9F&F,F&F,F;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "G uessing counts with Maple." }}{PARA 0 "" 0 "" {TEXT -1 232 "At this st age, we ask ourselves whether a closed form expression may exist. Fir st, let us examine the way we might proceed in a simple case, using Ma ple. Consider for instance the number# of NC-trees of size 30 and try t o factor it." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ifactor(coun t(nctree,size=30));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*>-%!G6#\"\"#\" \"$-F%6#F(F'-F%6#\"\"&\"\"\"-F%6#\"# " 0 "" {MPLTEXT 1 0 70 "seq([n,ifactor(count(nctree,size=n+1)/cou nt(nctree,size=n))],n=1..15);" }}{PARA 12 "" 1 "" {XPPMATH 20 "617$\" \"\"F$7$\"\"#-%!G6#\"\"$7$F**$-F(6#F&F&7$\"\"%**-F(6#\"\"&F$-F(6#\"#6F $F-!\"#F'!\"\"7$F4*,F'F$-F(6#\"\"(F$-F(6#\"#8F$F2F9F5F97$\"\"'*(F-F&-F (6#\"#*(F-F$-F(6#\"#>F$F " 0 "" {MPLTEXT 1 0 80 "seq([n,ifactor(n*(2*n+1)*count(nctree,size= n+1)/count(nctree,size=n))],n=1..15);" }}{PARA 12 "" 1 "" {XPPMATH 20 "617$\"\"\"-%!G6#\"\"$7$\"\"#*(-F&6#F*F$F%F$-F&6#\"\"&F$7$F(*(F,F*F%F$ -F&6#\"\"(F$7$\"\"%*(F%F$F.F$-F&6#\"#6F$7$F0*(F%F$F3F$-F&6#\"#8F$7$\" \"'*(F,F(F%F$-F&6#\"#F$7$\"\")*(F%F$F9F$ -F&6#\"#BF$7$\"\"**(F%F$F.F*F>F$7$\"#5**F,F$F%F$F3F$-F&6#\"#HF$7$F;*(F ,F7F%F$-F&6#\"#JF$7$\"#7**F%F$F.F$F3F$FDF$7$F@*(F%F$FIF$-F&6#\"#PF$7$ \"#9**F,F*F%F$F.F$-F&6#%\"#TF$7$\"#:**F,F$F%F$F9F$-F&6#\"#VF$" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "The numerators now seem to involve " }{XPPEDIT 18 0 "3*n-1" ",&*&\"\"$\"\"\"%\"nGF%F%\"\"\"!\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "3*m-2" ",&*&\"\"$\"\"\"%\"mGF%F%\" \"#!\"\"" }{TEXT -1 18 ". Thus we try next" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "seq([n,ifactor(n*(2*n+1)/(3*n-1)/(3*n-2)*count(nctree ,size=n+1)/count(nctree,size=n))],n=1..15);" }}{PARA 12 "" 1 "" {XPPMATH 20 "617$\"\"\"*&-%!G6#\"\"$F$-F'6#\"\"#!\"\"7$F,F%7$F)F%7$\" \"%F%7$\"\"&F%7$\"\"'F%7$\"\"(F%7$\"\")F%7$\"\"*F%7$\"#5F%7$\"#6F%7$\" #7F%7$\"#8F%7$\"#9F%7$\"#:F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "W e have thus found empirically evidence for the conjecture that the cou nts satisfy" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "u(n+1)=3/2*n*(2*n+1) /(3*n-1)/(3*n-2)*u(n);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6#,&%\"nG\"\"\"F)F),$*,F(F),&F(\"\"#F)F)F), &F(\"\"$!\"\"F)F0,&F(F/!\"#F)F0-F%6#F(F)#F/F-" }}}{EXCHG {PARA 0 "" 0 ""& {TEXT -1 49 "This can be solved by means of the Maple command " } {HYPERLNK 17 "rsolve" 2 "rsolve" "" }{TEXT -1 57 ", suggesting a close d binomial form for the coefficients." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "rsolve(\{\",u(1)=1\},u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*.\"\"$#\"\"\"\"\"#%#PiGF'-%&GAMMAG6#,$%\"nGF(F'-F+6# ,&F.F'#!\"\"F%F'F3-F+6#,&F.F'#!\"#F%F'F3)\"#7,$F.F3F'\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 81 "Naturally, we succeeded here only because the c ounts admit a direct product form." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "Guessing counts with Gfun. " }}{PARA 0 "" 0 "" {TEXT -1 60 "Our guessing task is greatly simplified if we appeal to the " }{HYPERLNK 17 "Gfun" 2 "gfun" "" }{TEXT -1 174 " package that determines automati cally recurrences for sequences and differential equations for the cor responding generating functions that match a given set of initial data ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "with(gfun);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7T%(LaplaceG%.algeb'raicsubsG%.algeqtodiffeqG %.algeqtoseriesG%.algfuntoalgeqG%&borelG%.cauchyproductG%.diffeq*diffe qG%.diffeq+diffeqG%,diffeqtorecG%)guesseqnG%(guessgfG%0hadamardproduct G%0holexprtodiffeqG%)invborelG%,listtoalgeqG%-listtodiffeqG%0listtohyp ergeomG%+listtolistG%.listtoratpolyG%*listtorecG%-listtoseriesG%5listt oseries/LaplaceG%1listtoseries/egfG%4listtoseries/lgdegfG%4listtoserie s/lgdogfG%1listtoseries/ogfG%4listtoseries/revegfG%4listtoseries/revog fG%,maxdegcoeffG%*maxdegeqnG%,maxordereqnG%,mindegcoeffG%*mindegeqnG%, minordereqnG%*optionsgfG%,poltodiffeqG%)poltorecG%/ratpolytocoeffG%(re c*recG%(rec+recG%,rectodiffeqG%*rectoprocG%.seriestoalgeqG%/seriestodi ffeqG%2seriestohypergeomG%-seriestolistG%0seriestoratpolyG%,seriestore cG%/seriestoseriesG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "First, we \+ build a list of small values:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "l: =[seq(count(nctree,size=n),n=0..20)];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"lG77\"\"!\"\"\"F'\"\"$\"#7\"#b\"$t#\"%G9\"%_x\"&jK%\"'vmC\"(:2(V \"\"(SYT)\")3r1]\"*s0$3I\"+?lwA=\",kcvC6\"\",f\\vG$o\"-N`aI?U\".lE/J'> E\"/+.HAHL;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The procedure " } {HYPERLNK 17 "gfun[listtorec]" 2 "gfun[listtorec]" "" }{TEXT -1 128 " \+ finds automatically a plausible recurrence with polynomial coefficient s, provided such a form exists; similarly, the procedure " }{HYPERLNK 17 "gfun[listtodiffeq]" 2 "gfun[listtodiffeq]" "" }{TEXT -1 113 " find s automatically a plausible differential equation with polynomial coef ficients, provided such a form exists." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "listtorec(l,u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$<%/-% \"uG6#\"\"\"F),&*&,(!\"'F)%\"nG\"#F*$F.\"\"#!#FF)-F'6#F.F)F)*&,&F.F1F0 \"\"%F)-F'6#,&F.F)F)F)F)F)/-F'6#\"\"!F>%$ogfG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "listtodiffeq(l,U(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$<%/-%\"UG6#\"\"!F),(*&,&!\"#\"\"\"%\"zG\"\"'F.-F'6#F/ F.F.*&F/F.-%%diffG6$F1F/F.\"\"#*&,&*$F/\"\"$\"#F*$F/F7!\"%F.-F56$F4F/F .F./--%\"DG6#F'F(F.%$ogfG" }}}{)EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "This candidate equation can be passed for possible solution to Maple's dso lve." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(op(1,\"),U(z));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"UG6#%\"zG*&F'\"\"\"-%*hypergeomG6 %7$#\"\"#\"\"$#F)F07##F0F/,$F'#\"#F\"\"%F)" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 34 "Proving solutions with Combstruct." }}{PARA 0 "" 0 "" {TEXT -1 133 "With the version of December 1996, we can find automatic ally a system of equations and in this particular case, an explicit so lution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "gfeqns(op(2..3,nc tree),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&/-%\"ZG6#%\"zGF(/-%\"FG F'*$,&\"\"\"F.-%\"BGF'!\"\"F1/F/*&F*\"\"#F%F./-%\"TGF'*&F%F.F*F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "nctree_sys:=gfsolve(op(2..3, nctree),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+nctree_sysG<&/-%\"ZG 6#%\"zGF*/-%\"BGF)*&-%'RootOfG6#,(%#_ZG!\"\"*&F3\"\"$F*\"\"\"F7F7F7\" \"#F*F7/-%\"TGF)*&F*F7F/F7/-%\"FGF)F/" }}}{EXCHG {PARA 0 "" 0* "" {TEXT -1 81 "This tells us in particular that the generating function \+ of non-crossing trees is" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "nctree_ gf:=subs(nctree_sys,B(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*nctre e_gfG*&-%'RootOfG6#,(%#_ZG!\"\"*&F*\"\"$%\"zG\"\"\"F/F/F/\"\"#F.F/" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "series(nctree_gf,z=0,12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+;%\"zG\"\"\"\"\"\"\"\"#\"\"#\"\"(\" \"$\"#I\"\"%\"$V\"\"\"&\"$G(\"\"'\"%wQ\"\"(\"&=8#\"\")\"'v,7\"\"*\"'!p !p\"#5\"(:?.%\"#6-%\"OG6#F%\"#7" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 264 10 "Conclusion" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "In particular, we have obtained automatically one of the theore ms of Noy (1994):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 7 "Theorem" }{TEXT -1 2 ". " }{TEXT 267 68 "The generating f unction of non crossing trees satisfies the equation" }}{PARA 264 "" 0 "" {XPPEDIT 18 0 "T(z)=1+z*T(z)^3" "/-%\"TG6#%\"zG,&\"\"\"\"\"\"*&F& F)*$-F$6#F&\"\"$F)F)" }{TE+XT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 25 "T he number of trees with " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 9 " \+ nodes is" }}{PARA 265 "" 0 "" {XPPEDIT 18 0 "binomial(3*n+1,n)/(3n+1) " "*&-%)binomialG6$,&*&\"\"$\"\"\"%\"nGF)F)\"\"\"F)F*F),&*&\"\"$F)F*F) F)\"\"\"F)!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "The result of " }{HYPERLNK 17 "combstruct [gfsolve]" 2 "combstruct[gfsolve]" "" }{TEXT -1 72 " gives a valid pro of of the generating function equation. The result of " }{HYPERLNK 17 "gfun[guessgf]" 2 "gfun[guessgf]" "" }{TEXT -1 13 " followed by " } {HYPERLNK 17 "gfun[listtodiffeq]" 2 "gfun[listtodiffeq]" "" }{TEXT -1 166 " is a priori only heuristic. However, once a candidate recurrence or differential equation has been found, it may then rigorously be ch ecked from withing gfun itself." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "First, we compute automatically a differential equation from the g enerating function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1, 0 30 "algf untoalgeq(nctree_gf,Y(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%\"zG! \"\"%\"YG\"\"\"*$F&\"\"#!\"#*$F&\"\"$F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "algeqtodiffeq(\",Y(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*!\"#\"\"\"-%\"YG6#%\"zG\"\"$*&,&\"\"#F%F)!#FF%-%%diffG6$F&F)F% F%*&,&*$F)F-F.F)\"\"%F%-F06$F/F)F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 165 "This result has proven value and it coincides with the differe ntial equation that was previously \"guessed\". Thus, the theorem has \+ been established \"automatically\"..." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "Comtet's counting of slicings. " }}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 8 "Problem." }}{PARA 0 "" 0 "" {TEXT 258 6 "Given " } {XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT 268 78 " points on a circle, count the number of slicings, that is to say, graphs cons" }{TEXT 259 10 "i sting of " }{XPPEDIT 18 0 "floor(n/2)" "-%&floorG6#*&%\"nG\"\"\"\"\"#! \"\"" }{TEXT 269 47 " edges that do not intersect inside the circle." }}}{SECT 1 {PARA 4 "" -0 "" {TEXT -1 47 "Specification, random generati on, and counting." }}{PARA 0 "" 0 "" {TEXT -1 247 "Consider first the \+ case of an even number of vertices. A slicing decomposes into a centra l edge (initially, the edge that contains vertex 0) and into a left an d right slicing that are each of a similar nature. This is readily spe cified as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "sl0:= \{Edge=Prod(Z,Z),S=Union(Epsilon,Prod(S,Edge,S))\}: slicing:=[S,sl0,un labelled]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "seq(count(sli cing,size=n),n=0..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "67\"\"\"\"\"!F #F$\"\"#F$\"\"&F$\"#9F$\"#UF$\"$K\"F$\"$H%F$\"%I9F$\"%i[F$\"&'z;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Our old friends, the Catalan numbe rs strike again." }}{PARA 0 "" 0 "" {TEXT -1 237 "An interest of the m ethod is to allow for the counting of odd slicings, where one has a fr ee unattached point. The following specification just says that the \" free point\" lies somewhere either left or right. or at the root of the slicing." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "sl1:=sl0 unio n \{Free=Z, S1=Union(Prod(S1,Edge,S),Prod(S,Edge,S1),Prod(S,Free,S))\} : slicing1:=[S1,sl1,unlabelled]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "seq(count(slicing1,size=n),n=0..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "67\"\"!\"\"\"F#\"\"%F#\"#:F#\"#cF#\"$5#F#\"$#zF#\"%. IF#\"&S9\"F#\"&eP%F#\"'gz;F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "draw(slicing1,size=20);" }}{PARA 8 "" 1 "" {TEXT -1 69 "Error, ( in combstruct/drawgrammar) there is no structure of this size" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "There, the system has recognized \+ the absence of any structure of even size. Everything goes well with a n odd size." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "draw(slicing 1,size=19);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%%ProdG6%%(EpsilonG-F$ 6$%\"ZGF)-F$6%-F$6%-F$6%-F$6%F&F'-F$6%-F$6%F&F'F&F'F4F'F&F)F&F'-F$6%F& F'F4" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 51 "Jordan's couting of no n-crossing Hami/ltonian paths." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "P roblem." }}{PARA 0 "" 0 "" {TEXT 260 6 "Given " }{XPPEDIT 18 0 "n" "I \"nG6\"" }{TEXT 270 90 " points on a circle, determine the number of h amiltonian paths that have no crossing edges" }{TEXT -1 1 "." }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "Specification, random generation \+ and couting." }}{PARA 0 "" 0 "" {TEXT -1 237 "We discuss here the simp ler problem of specifying non-crossing hamiltonian paths that start at the designated root node of index 0. An initial segment of such a pat h can be continued in two ways, either by moving to the \"next\" point (an " }{XPPEDIT 18 0 "N" "I\"NG6\"" }{TEXT -1 72 "-move) away from th e root, or by crossing over to the opposite side (an " }{XPPEDIT 18 0 "X" "I\"XG6\"" }{TEXT -1 28 "-move). The final step (the " }{XPPEDIT 18 0 "F" "I\"FG6\"" }{TEXT -1 70 "-step) is fully determined by the co ntext. Thus, the specification is:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "ham_path:=[H,\{H=Prod(Sequence(Union0(N,X)),F),F=Z,N=Z ,X=Z\},unlabelled]:" }}}{PARA 0 "" 0 "" {TEXT -1 34 "We count, draw, e tc. For instance:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "seq(cou nt(ham_path,size=n),n=0..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "67\"\"! \"\"\"\"\"#\"\"%\"\")\"#;\"#K\"#k\"$G\"\"$c#\"$7&\"%C5\"%[?\"%'4%\"%#> )\"&%Q;\"&oF$\"&Ob'\"'s58\"'W@E\"')GC&" }}}{PARA 0 "" 0 "" {TEXT -1 134 "Naturally, the result is the powers of two. We leave it as an exe rcise to prove that the number of unconstrained hamiltonian paths is \+ " }{XPPEDIT 18 0 "n*2^(n-3)" "*&%\"nG\"\"\")\"\"#,&F#F$\"\"$!\"\"F$" } {TEXT -1 161 ". The idea is to decompose such paths as either starting right, or left, or as a gluing (with suitable constraints) of two pat hs that emenate from the root node." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 34 "Other non-crossing configurations." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Discussion." }}{PARA 0 "" 0 "" {TEXT -1 295 "By the \+ same principles, it is possible to enumerate, list exhautively, draw a t random, et1c, a large number of geometrical configurations. Examples \+ are general non-crossing graphs, forests, dissections, trees and graph s of bounded degree (Noy , September1996). We discuss here the case of forest." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Problem." }}{PARA 0 " " 0 "" {TEXT 261 67 "How many forests of trees with no crossing edges \+ can be built from " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT 271 20 " poin ts on a circle?" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Specification , counting and generating functions." }}{PARA 0 "" 0 "" {TEXT -1 255 " A forest has a skeletton that is defined as the non-crossing tree that contains the vertex of smallest index. A (possibly empty) forest is t hen attached to each node of the skeleton. Thus forests are recursivel y definable as non-crossing trees of forests." }}{PARA 0 "" 0 "" {TEXT -1 70 "The corresponding specification is thus derived from that of NC-trees." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "forest:=[F, \{F=Union(Epsilon,T),T=Prod(Z1,P),P=Seq2uence(B),B=Prod(P,Z1,P),Z1=Prod (Z,F)\},unlabelled]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "seq (count(forest,size=n),n=0..20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "67\" \"\"F#\"\"#\"\"(\"#L\"$\"=\"%$3\"\"%ao\"&6^%\"'HcI\"($G<@\")7#H\\\"\"* +0z1\"\"*-c.t(\"+BdF_c\",@F\"RoT\"-fjL\"p4$\".&[_Dx:B\"/rLf&R:u\"\"0W( o]LK;8\"0GQT\\MU***" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "draw (forest,size=20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%%ProdG6$-F$6$% \"ZG-F$6$-F$6$F(%(EpsilonGF--%)SequenceG6$-F$6%F-F+F--F$6%F--F$6$F(-F$ 6$F&F--F/6#-F$6%-F/6#-F$6%F-F+-F/6'F1-F$6%-F/6$-F$6%-F/6#F1F+F-F1F+F-F 1-F$6%-F/6#-F$6%-F/6$F1F1F+FIF+F-F1F+F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "forest_sys:=gfsolve(op(2..3,forest),z);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%+forest_sysG<(/-%\"ZG6#%\"zGF*/-%#Z1GF),$*&F* \"\"\",&*&-%'RootOfG6#,**&%#_ZG\"\"$F*F0F0*&F8\"\"#F*F0F0*&,&!\"\"F0F* F>F0F8F0F0F0F0F0F*F0F0F>F0F>F>/-%\"FGF),$*$F1F>F>/-%\"BGF),$*(F3F;F*F0 F1F>F>/-%\"PGF)F3/-%\"TGF),$*(F*F0F1F>F3F0F>" }}}{EXCHG {PARA 0 "> " 3 0 "" {MPLTEXT 1 0 29 "evala(subs(forest_sys,F(z)));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,,*&-%'RootOfG6#,**&%#_ZG\"\"$%\"zG\"\"\"F-*&F*\"\"#F ,F-F-*&,&!\"\"F-F,F2F-F*F-F-F-F-F/F,F/F-*&F%F-F,F/F-*&F%F-F,F-F-*$F,F/ F2F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "We automatically obtain a theorem of Noy (1996)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "series(\",z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"zG\"\"\"\"\" !F%\"\"\"\"\"#\"\"#\"\"(\"\"$\"#L\"\"%\"$\"=\"\"&-%\"OG6#F%\"\"'" }}}} }{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Conclusions." }}{PARA 0 "" 0 "" {TEXT -1 29 "We have demonstrated the way " }{TEXT 272 10 "Combstruct " }{TEXT -1 67 " can be used to do experimental combinatorics. In conj unction with " }{TEXT 273 4 "Gfun" }{TEXT -1 97 ", it can help generat e conjectures and even derive automatic proofs of theorems in combinat orics." }}}}{MARK "1 0 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 } F>F>/-%\"FGF),$*$F1F>F>/-%\"BGF),$*(F3F;F*F0 F1F>F>/-%\"PGF)F3/-%\"TGF),$*(F*F0F1F>F3F0F>" }}}{EXCHG {PARA 0 "> " 3 invlaplac¿ invo ¶Jinvok½Œ Aý\involutîxinvolvs*à¿ ç £ µ%'3(ú+ ,½4W5˜@¶J*O†P¼QWVÌ^jÎjBk9q!vkwîx:yinzW5io¿ Õ\ ÎjBkëlioaefduÕ\ioazÕ\iobëliobfëlioc Š W5ioeÕ\iofÕ\iohÎjiolw¿ ion{.àŠ ç ‚ £ 9"'3(½8Â;-B”E.F¼Q‰\Õ\ý\à^Ü_ÎjëlÔmÊo9q•rt:yional*OBk:yiong Õ\:yions+ i£ 9‡/ A¼Q ^jÎj:yioriÕ\iousÎjiowÕ\ioyëlip ¿ Õ\Îjëlipffqfhpfdpfhq:yipfiipÕ\ipl‰\ Ü_Bk Š èK-B¼Q-[Îj:yàA»M*O‚Qý\djÎjBkŸp!v:yëlÔmŸp9q•r sòs !vîx:y€ uskb¿ usly¿ ussëlustà”EÕ\usu ,usual/àŠ ¿ P Í ‡/¶JÕ\ ^Îj:yut ¿ ç 7.FÎjo:yutaÕ\utat '»Mutation †Pý\ute¿ $@‰\ÊoutedBkutesJ@*O¼QutfÎjuthkÎjution¿ ¼QoutkfëlutlÕ\utoutocomb!v utocomplet 3q8 utodeterminizË-‰\Ìdecalphaunixmaplinputcourimathtimehyperlinkcommoutputlucidanormaheadtimesattributgrammarcombinatoricwayexpresrecursivedefinpropertstructurheregeneratcombstructcombstructtypicalexamplpathlengthtreenumbcyclpermutatevensizeitselffunctionintroducworksheetofferthuserdescribautomaticallyinformataboutrelatmultivariatcanextractaveragvaluesotherstatisticmorecontextpleasreadcomplexitanalysiavailablathttpalgoinriafrmishnainfomatcheckhelppageeginfireupwithingtributaccompanyrammarbasedsubatomtakeconstantbinarinternalipltotalsumdistanceachnoderootthisumssubtreplustheirtreeunionprodnpathinterpretatsyntaxfollowgiventypecaseformasechildrenplusadditnotatimplproductoverallcomponentpredefinattributespecificatattributehighdependantuponthusspecifymirrorthemslvesiteratoperatorsuchsetsequencefunctmemberefinitnonplancorresponddefinitsumsgrammarnptpatmustlinearß ÀDãdecalphaunixmaplinputcourimathtimehyperlinkoutputhelpnormalnormalheadbulletitemregexpcountautomarkovtransformbernoulliautomatonmarkovusagcallsequencautoorderlettparametergrammarspecificatderchaintransitinfodescriptfunctreturnequivalntbothautomatadescribtableinternalspecificationcomplettransitformetterbelongoriginalalphabetatmostequalsourcplusstatmeanletterhavereadbeforentereadgoexamplwithregexpcounrecognizregprodatomntautoregexptoautoprinttautotablegprunedgtrueginitgagtransgliststatesgbgfffinalgfhalphabetgintomarkovtmarkovautokgfbliststatesgfinalgfseealsoalsoregexpcountautotogramautotogramgramtoautogrammarkovgrammarkovkudhÕ\kueÕ\kuezëlkuk¿ kummermdkummerudkuoaÕ\kurjëlkutta¿ kuwgg¿ kvŠ Õ\Îjkvl"kvlef"kvlfd¿ kvlmf"kvmÕ\kwŠ ¿ Õ\ëlkwbBkkwbdjÎjkwesÕ\kwfÕ\kwlÕ\kwpëlkwsbnëlkwwÕ\kx¿ Õ\ÎjkxbÕ\kxdnÕ\kxdnrÕ\kxkëlkxvÕ\kxyÕ\ky¿ Õ\ëloyf Õ\ëloyfduÕ\oyfdufÕ\oyfvàoyhboëloyl W5ëloymkÎjoyr¿ oyrxÕ\oysvgëloyw¿ oyyëloz'à¿ "ú+.F†PÕ\Îjël ozahho¿ ozbknfduÕ\ozdòsozlëlozoëloztëlozuW5pa;*¿ P "Ë-V1W5˜@†PÕ\ Ü_Îjëlòskwpabilit òs!vpacP Œ7packP packaÍ R67packagÍÄàë ¿ P 9½CŒÆì?è¯K¿Í µ%'3(ú+ ,‡/ V1½4W5R67S:ˆ;Â;ü;<¥=ó=J@ AàA-B.F»I¶J»MCN*O†P!Q‚QWVXZÕ\ý\å] ^Ü_d†ijÎjÊo9qÁq•r!vkwîx:yXZ-[Õ\ý\å] ^Ì^à^_Ü_)``§b|cd*džd†iÎjBkëlÔmypŸp9qÁq sòs!vkwuybutanW5bution P îxbuttòsbutterfÕ\òsbutterfl Õ\òs butterfliÕ\bvŠ ¿ "†PÕ\ëlbvd¿ bvklëlbvlÕ\bvsW5bvtoxÕ\bvuxÕ\bvytfëlbwÕ\ÎjëlbwbcvzkzëlbwdÕ\bwgrÎjbwhv¿ bwjxiÕ\bwm¿ bwsëlbwwëlbx¿ W5Õ\ëlbxclëlbxe Õ\ëlbxfÎj:ymdf"mdfdimÎjmdgòsmdjëlü;$@¼Q‰\Õ\ ý\å] ^dÎjBkëloÊoòs!v:ycoalŠ ken àŠ è-B¼QÎj:yÊo zvasbumffdmÕ\zveëlzveoxëlzvhàzvjëlzvkëlzvot¿ zvrÕ\zvslÕ\zvxëlzvywëlzvzŠ zvzvjÕ\zw+àŠ ¿ k"Õ\ÎjBkëlzwaa¿ zwcvëlzwdëlzwdfëlzwf†PzwgozwicxÎjzwjëlzwnÕ\zwpëlzwrjëlzwxiehÕ\zwz¿ zx ¿ "W5Õ\ël:yzxehëlzxfëlzxkstÕ\zxlëlzxp¿ zxtëlzxuzëlzy à¿ W5Õ\ÎjëlzyfÕ\zym ¿ Îjzymc¿ zypëlzyzëlzz ¿ "Õ\ÎjëlzzaÕ\zzctÕ\zzjÕ\zzpkÕ\zzuëlzzyxÕ\WVibmintellinuxmaplinputcourimathtimehyperlinkcommoutputhelpheadnormalbulletitemfunctholonomhypergeomdfinitcomputfirstorderequationsatisfisequencusagcallfgagparameteralgebraicexpressorealgebrasynopsidescriptcommandsatisfiedhypergeometrichyperexponentialcurrentnowayautomaticallyobtainsystemoperatornonfunctionequencfuturwithintroductcommanpackagmgfunmayanypossibinvolvqgcalculuhowevypergeomonlymeantconvenitoolreturnedvaluundeterminequatexistcaseeachskewunctpartcanusedafterperformholonomyalwayaccesslongformexamplnwithdiffdxdymunhypergeomexpnudygnugdxgfmugfshiftsnskbinomialskgkgfngsngfhypergeometricmgfunnaqshiftqfactorialhgkgqfactorialgngfqgfskgfseealsoalso[ configurat\exhaustä gaussagmg‹ idwcxŒ leFogfgGregular< slh= tuneBtoseries] - convert a list into a series " } } {PARA 0 "> " 0 "" {TEXT 26 10 "Function: " } {TEXT -1 49 "gfun[seriestolist] - convert a series into a list" } } {PARA 0 "> " 0 "" {TEXT 26 10 "Function: " } {TEXT -1 45 "gfun[listtolist] - convert a list into a list" } } {PARA 0 "> " 0 "" {TEXT 26 10 "Function: " } {TEXT -1 53 "gfun[seriestoseries] - convert a series into a series" } } {PARA 0 "> " 0 "usage" {TEXT 26 17 "Calling Sequence:" } {TEXT -1 95 "\n listtoseries(l, x, gf)\n listtolist(l, gf)\n seriestolist(s, gf)\n seriestoseries(s, gf)" } } {PARA 0 "> " 0 "" {TEXT 26 11 "Parameters:" } {TEXT -1 4 "\n " } {TEXT 23 7 "l - " } {TEXT -1 10 "a list\n " } {TEXT 23 7 "s - " } {TEXT -1 12 "a series\n " } {TEXT 23 7 "x - " } {TEXT -1 10 "a name\n " } {TEXT 23 7 "[gf] - " } {TEXT -1 10 "(optional)" } } } {SECT 0 {PARA 0 "> " 0 "synopsis" {TEXT 26 12 "DescriptdsyÎjdt Õ\Îjëldtc àŠ dtf"dthkng:ydtl"dto ¿ ÎjdtontqŠ dtu¿ du¿ ‡/†PÕ\Îjdua.Fdual#MÍ ‡/ˆ;.F%_Êo9q:ydualizat.Fduce ¼Q ^ductàA»I ^9q ducwrwfduÕ\dudëldudhÕ\due3Š ¿ £ µ%J@†PÕ\Bkoòs:ydufÕ\ dufeuffgmÕ\dug‡/dugf‡/duj"dule ?ëlduluÕ\dumm‡/duo¼Qduplicat "òsduqÕ\dure ÆÜ_duringi½8kwdutour•rdv¿ Õ\ëldvbÕ\dvbfdcvhëldvcu†PdvloëldvmfÕ\dvvÎjdvzpcëldw ¿ Õ\dwe¿ dwf†Pdwgsprëlý\Âdecalphaunixmaplinputcourimathtimehyperlinkoutputnormalheadingbulletitemfixedwidthpredefinstructurcombstructpackagusagcallsequencfunctstructargssizefunctionstructparameterlistavailablargumntcorrespondnonnegatintegspecifyobjectstringallsizwhenchosenallpossiblgiveneachdefaultbehaviorinfodescriptsomespecialcombinatorialclasspredefintheycombinatsubsetcombinatelementsetcasentegtreatnpermutatpermutatelementnumbnpartitionpartitpositintegerintosumswithoutregardorderncompositcompositintegerpartatleastsummandsimportantdefaultdiffernamesametructuryoumayyourownwillunderstoodfunctioncreatefoocreatprocedurcountdrawallstructiterstructthesmusttaketheirfirstallsizesfollowwhatevargumentstructurneedthusinvokrawexamplwithcombstrucsizeusingcountiteratoroverpermutationconsistonlyanyizeseealsoalsoallstructsombstruct configuratioÕ\confirm¿ Îjël:yconflict½Œ A:yconformationalW5congraphòsconj conjectur# à¿ K¼QBko:yconjunct Îj!vconn¿ ÎjBkconnect#7¿ Õ\ÎjBk òs !vîxconnectiÎj connectiong¿  connectivitŠ Îj!v connectvitŠ cons# Œ'3(˜@˜]òs:yconsecutŠ Œ†P¼Qòs!v:y consecutivÕ\ consecutive¼Qconseqòsconsiconsidg2à¿ 9Æ?"'3(‡/W5J@†P!Q¼Q‰\Ü_jBkëlòs!v:yconsider+ûèÍ "µ%Bkëloòs:y considerablŠ  considerat?µ%ú+½4consistKàŠ ¿ P Æ"'J@CN*OÕ\ý\Ü_ÎjBkoòsîxgfeqnsg+à:'W5 A¼QÜ_Îj:y laguerreldlahÕ\lamjlambaÜ_j•rlambda Ü_jo•rlambdagÜ_o•rlambertàlambertwà lambertwg à$@à^lang ël:ylangu ¼QlanguagW33i‚ ½{w"Ë-/78q8LD˜]jÎjëltîx:yŠ}lapl µ%àAlaplac à¿ K S-[:ylaplaceg ¿ †PÎj:ylariw"ˆ;:ylarat*OlareÊolargKàŠ ¿ ÖJ@†P¼Q1Vžd†iÎjBkëloòs!v:ylarge Ü_olaser¼Qlass¿ lassical£ last;Š £ P µ%J@†P¼Q‰\Õ\Ü_ëlŸpòslation¿ latt' iÆW5ü;J@Õ\ÎjòslatticŠ ¿ !vlaurencîx lauricella ^lawà†P:ynlcjëlnleadmonXZnleaf:nlifëlnlik'nlistÎj nlisttoalgeq¤N nlisttodiffeqX9 nlisttorecà nlisttoser Snll"nllbuëlnlm†Pnlo¿ nlocal ÆÕ\Ü_jBkîxnlog˜]nly †Pdnm#9:"˜@¼QÕ\Îjîx nmarkleaf:nmax¿ nmaxg¿ nmbf:ynmbintre :"nmdòsnmeW5nmehzptiëlnmember'nmgLDnmij†Pnmm Š ¿ nmmm ¿ òs:ykalÎjkaneW5kapµ%kappa ¿ µ%kappag ¿ µ%kappagfµ%kaudlëlkavbwëlkawuŠ kb†PÕ\ÎjëlkbbyÕ\kbfÕ\kbhfduÕ\kbpëlkbrgëlkbts:ykbwÕ\kbxlydŠ kbz¿ kc¿ ú+Õ\ëlkca¿ kcdgffëlkcu¼Qkcvckczdëlkd "W5Õ\ëlkdaŠ kdaofÕ\kdd¿ kdgÕ\kdibkÕ\ neighbour àòs neighbourhoodk¿J@ãMÔm sneithC˜@Ì^nek"nelÊonelsÕ\nenÕ\nendG/àŠ ¿ Ƈ/W5½8J@¶J†PÕ\ Ü_jÎjBkîxnentialJ@neq*dnequival sner# ?‡/¶JXZå] ^Ì^kwneral˜@ neralizat:ynerat#Œ'Â;”EBkëlÔmneratorCnerfypnericW5nest‚ netëlnevalf àòsnever¿ ½4àA†P‰\gedƒ4 gegenbauergdgeildÕ\gelulpëlgemtÎjgen' ‚ ¿W5†PÕ\Ü_ëlÔm:y\nageg:y iÆW5ü;J@Õ\ÎjòslatticŠ ¿ !vlaurencîx lauricella ^lawà†P:ynatu ¿ 3(natur †Pîx:youtsidµ%Îjkwouusëloux¿ ouyÎjouzyhoÕ\ov'i¿ 9"ÀDÕ\ël tîxovautogtoveë ¿ Æ-B¶Jëlover§Mà.ë ‚ £ Æ?¯Í "3( ú+‡/½4$@J@àA-B¶J*O!Q‚Q¼Q‰\ý\ ^Ì^Ü_d†ijÎjBkël9qòs!v:yoveral P 'overcomòsoverlap Š ½ïLDëltoverlappŠ LDëloverload.overriddS: overriddenÂ;overview 9'ovf Õ\ëlovgafzufëlovhÕ\ovjÕ\ovjkk¿ ovlÕ\ovlcmdÕ\ovn ¼QëlovncfÕ\ovnrxëlovoaÕ\ovsëlovwÕ\owŠ ¿ "W5àAÕ\ëlowaëlŠ èK-B¼Q-[Îj:yevic:yrsivç rskrÎjrslvÕ\rsmÕ\rso¿ rsol µ%J@rsolg µ%J@rsolv#+" µ%[,)`Bk oòsrsolveg"Bkòsrsprëlrssz¿ rstàArstruct Arsvdëlrt'¿ Ö†PÕ\ ÎjBkël•rkwrtakëlrtdimCrtdsëlrtenàArtg ‡/‰\rtiëlrticalŠ rtie àÎjrtjxffdmÕ\ rtjxffdmfÕ\rtncëlrtniëlrturbat£ rtvjxtëlrtzŠ ru#¿ "W5R6Õ\ ël9q:yrucial!vruct' ½:{ A‰\jBkëlŸpructsg¼Qructur V1ý\rudhÕ\Õ\lfe ¿ Õ\lfejlÕ\lfexf:ylffÕ\lffanÕ\lffcmÕ\lffeÕ\îx:y implementa9q implementatà3(X9-B¤NÕ\Bk implementati½8impli ÆÜ_implicÕ\implicit࣠P W5J@†P implicitplotÕ\implif ‚ Îj importanc$@ important:7Â;Õ\ý\impos:yimpossib¼Qimproj improbablëlimprovjimprovem†iimptfŠ imtf¿ imvfÕ\imwÕ\imyfd¿ imyfhgq¿ imzëlinal¼QªYinar•rinatJ@ý\ ^j inatorialP ' AëlinatoricP !v:y inbetweenÕ\inc¼QincarnatÜ_j•r)`ÎjëlÔmo sòst!vkw :yuy@˜@àA-BCN!Q‚Q ¼QÕ\˜] ^dÎjBk9qÁq•r!v:yŠ}½4†P¼QÕ\Îjëlekfëlekl ¿ ëlekmÕ\ekrÕ\ekrih¿ eksraW5ekvÎjekvfÕ\ekxyÕ\el36àŠ ë ¿ "3(ˆ;Õ\å]ëlÊo:yeldcëlelds¿ eleý\electroÜ_ electronicjelegantŠ elemk-àÆ½'3(½8X9ü;<ó=8@˜@àA¤N‚QÕ\Ü_)`§bždBkÔmÊoîx:yuyelemen3(elements=à.£ C?ï¿'‡/R6<ó=˜@»MãM*O¼Q SXZý\)`§bŸp!vkwîx:y elementar˜@Õ\dòs:yelevatŠ 'W5 A¼QÜ_Îj:yfinit³z.ë ‚ P ½ìè¯¿Í w"3(ú+Ë-/‡/½4Þ5R6778q8J@˜@àA-BCN!Q‚Q ¼QÕ\˜] ^dÎjBk9qÁq•r!v:yŠ}Ë-tatorialÆW5Bkdevic:ypartg Õ\òspartggÕ\parti'partialŠ †Pd:yparticu ˆ;:y particula" particular“9.Š ¿ P ½ŒÆ:KÍ "µ%'Ð0W5ü;¥= AàA†P‚Q¼Q SÕ\ ^Ì^Ü_jBkëlÊoòskwîx:ypartit/8£ "'W5˜@ A¼Q Õ\ ý\•ròs partitioòs partitional˜@partsgÕ\partssyÕ\ partssysgÕ\pascal¼Qpass+ ࣠:è†P¼QÕ\:yuypat‰\Îjëlpath;e¿ ç "Ë-‰\Ü_jÎjBkòs!v:ypathl"Ü_òspathlenj pathlength(ç "‰\Ü_ j !v pathlengthgÜ_ pathlengthgf Ü_jpation†Ppatra Îj:ypatt ël:ypattern?~3àiŠ ¿ ˜@†P¼QÎjëlWt!v:y availaiblÎjaveòsaverjaveraBkaveragKYàiŠ ç £ :"$@”E†P‰\Ü_jBk•ròs !vîxaves "òsavg ‰\òsavgg ‰\òsavi¿ avk¿ avnbdŠ avnbdgŠ avnf.Favnx¿ avoÎjavoiÎjavoid#)¿ <J@”EÎjòs!v:yavutÕ\awW5.F†Pëlawa½4awaraway ëlawingW5awmëlawvëlawxo¿ ax¿ Õ\ëlaxes¿ Õ\:yaxesl¿ axeslab Õ\:y axeslabelsg à¿ "Õ\Îj axesstyleg"Õ\:yaxhcël½8J@˜@àA-B¼QÕ\dÎjBkëloòsevala ú+evalb‡/½8†PBkll "Îjflll"flnŠ  !vîx:y€ efpmyrcufdffdbffofvfdfvetfcopffvfzyfjqvqfbrxxefzgafjrfjqqfjrfjrxpffhstfjrqfhtfnpwcubuxefqffffvfjwdecomposcomponentrootuniquefactontainedgewithsmallestnumberinitialleftrightsubtriangulattheirpositrespectfirstloadcombstructallstructsgcountgrawgfinishedggfeqnsggfseriesggfsolvegiterstructsgnextstructgprogwritatomicsymbolzgdenottrainglarbitrartriangulatmustcarefulisolatsidedwhereithemptthushegrammarunionprodepsilonunlabelledcleareminiscbinarsearchtreenowthsizecouplsecondsetonceallcomplettablequalfuqqrofirstfewvaluseqyehuswnqwnnzwqwzjswklknowncatalanhonorfrenchbelgianmathematicianoutmainproptiesrandomdrawatamongstthosnygivennoteuniformitgrantpriorialgorithmscontainnaturalformatdictatsizesaycompriscorresponddodecagonprodgzgfepsilongfobjectstandardconstructvisualisprocedurlistprocconvertmapendorgedgelocalseelifelseopfinendplottcossinexpLanddgespicolorblueaxesnonepjzfbffqfkfcfflfrffrflffhnfzfzfcofofhnfdoffipfcfrffeqfjpqfeqfeqofeqflfffrfeqsffrfepfepfdofdsfkfblourgexhaustdecembreleasalsoexhautivenfiguratprovusefultestinsmallcasethoughphysicallimitatowingexponentialgrowthtrallstructepsilongfaffeffoundequationagainusesformassociatfollowgfeqnunlabelltgfgfsygfsolvtrianggfsysgsolutobtainexplicitcorrectneschecktaylorexpanscomparcombstrucgiveusseriogalternativefunctiongfserevennoclosedavailablfsertablegcrosscalluiltverticassumviewnodeconnectothereachmgconnectedbutterfnoncrossinattachanglebracketgletcrossingncfgforestprunbgnctresequencjkvmcsytlwakcvcvgailehahlsequenceguessageaskourselvwhethclosexpressexistexaminwaymightprocemapleinstanctryfactorifactorcounjfvfhfnfrftfzfobservhighcompositinvolvprimrsbetweenstrongindicatnvolvfactorialbinomialcoefficientpatternguesstakingsuccessquMotientcountfpffafzffinfratioinvolvengfdenominatorleastwhenevfifnumeratorseemmgfnextempiricalevidencconjecturntssatisfugcommandrsolvsuggestpigfgammagsucceedonlybecausountadmitdirectproductourtaskgreatsimplifiappealdeterminautomatidifferentialcorrespondfunctionmatchdatalaplacegalgebraicsubsgalgeqtodiffeqgalgeqtoseriesgalgfuntoalgeqgborelgcauchyproductgdiffeqdiffqgdiffeqgdiffeqtorecgguesseqngguessgfghadamardproductholexprtodiffeqginvborelglisttoalgeqglisttodiffeqglisttohypergeomglisttolistglisttoratpolyglisttorecglisttoseriesglisttoserlisttoseregfglgdegfglisttoserilgdogfgogfgrevegfgrevogmaxdegcoeffgmaxdegeqngmaxordereqngmindegcoeffgmindegeqngminordereqngoptionsgfgpoltodiffeqgpoltorecgratpolytocoeffgrerecgrecrectodiffeqgrectoprocgseriestoalgeqgseriestodiffeqgseriestohypergeomgseriestolistgseriestoratpolygseriestorcgseriestoseriesgbuildlglisttorecplausiblpolynomialcoefficiprovidsimilarlisttodifNfeqequatcoefficientffdiffgdgcandidatpassdsodsolvhypergeomgsolutionautomaticallylutionbgfsyssysgrootofgfgftellgfsubsnctrgfgwqconclustheormstheoremunctbinomialgvalidproguessgfheuristichowevrigorouscheckeditselfcomputeneratalgfuntoalgeqalgeqtodiffeqprovencoinciddifferntialpreviousestablishslicinggraphconsstingfloorfloorgdoinsidgeneratislicdecomposcentravertexnaturreadispecifislunlabellslicingufoldfriendstrikinterestethodallowoddunattachjustsaysfreeliessomewheruniocfepgzdrawgrammarstructurrecognabsenceverythgoescouthamiltonianpathroblemamiltoniandiscussimplerspecifystartdesignatindexsegmpatcontinumovingmoveawayoveroppositsidexgfinalstepfullntexthamtcobpowerleavexercisunconstrainideagluingsuitablconstrainthsemenatsameprincipletclarggeometricalgeneraldissectbounddegrehowbuiltpoinskelettonpossibhenskeletonrecursiveldefinablaohcibdfrotfjldxrlflkgqtmuOpgfevalademonstratusedconjhelpproofcombinatoricvhypergeomgsolutionautomaticallylutionbgfsyssysgrootofgfgftellgfsubsnctrgfgwqconclustheormstheoremunctbinomialgvalidproguessgfheuristichowevrigorouscheckeditselfcomputeneratalgfuntoalgeqalgeqtodiffeqprovencoinciddifferntialpreviousestablishslicinggraphconsstingfloorfloorgdoinsidgeneratislicdecomposcentravertexnaturreadispecifislunlabellslicingufoldfriendstrikinterestethodallowoddunattachjustsaysfreeliessomewheruniocfepgzdrawgrammarstructurrecognabsenceverythgoescouthamiltonianpathroblemamiltoniandiscussimplerspecifystartdesignatindexsegmpatcontinumovingmoveawayoveroppositsidexgfinalstepfullntexthamtcobpowerleavexercisunconstrainideagluingsuitablconstrainthsemenatsameprincipletclarggeometricalgeneraldissectbounddegrehowbuiltpoinskelettonpossibhenskeletonrecursiveldefinablaohcibdfrotfjldxrlflkgqtmuO†PèŽ{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 " " 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 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 "" 1 18 0 0 0 0 0 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 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{QPSTYLE "T ext Output" -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 "" 2 6 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 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 "" 1 16 0 0 0 0 0 1 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 ""R -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 siSm 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) exploitT 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\"\"\"U\"\"\"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'!\"\"-%$SumG6$*&V-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 "" 0W "" {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 "ThXis 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%1listtoserYies/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%$ogfG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "The recurr ence transforms into a differential equation by means of " }{HYPERLNK 17 "gfun[rectodiffeq]" 2 "gfun[rectodiffeq]" "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ode:=rectodiffeq(op(1,rec),u(n),Y(z));" }}{PARA 11 " " 1 "" Z{XPPMATH 20 "6#>%$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#aFbo#\"#]\"\"*/-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$F`pFWFinFin" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Note that a simple rational solution is detected by dsolve. This entba ils a reduction of order," }}{PARA 0 "" 0 "" {TEXT -1 92 "and a comple te algorithm exists for order 2 (in fact that Maple succeeds in bypass ing here)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "infolevel[dsolve]:=5: M2_z:=factor(op(2,dsolve(ode2,Y(z))));" }}{PARA 6 "" 1 "" {TEXT -1 421 "dsolve/diffeq/polylinearODE: checking Euler equation\ndsolve/di ffeq/expsols: trying exponential solutions\ndsolve/diffeq/expsols: \+ rational solutions partially successful. Result(s)= (1-3*z)/(-1+z)^ 3\ndsolve/diffeq/expsols_solvericcati: all solutions by polynomial p art\ndsolve/diffeq/expsols: expon. solutions partially successful. R esult(s) = exp(Int((-2*z^2+z-2)/(z^2-z),z)), exp(Int((-4*z^2-2*z)/(- 1+z^2),z))" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%M2_zG,$*&,,\"\"\"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 lceads 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$\"?'>Q1a6)3a-im;p%\\&F'\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "evalf(subs(n=50,var_n_asympt),30); evalf(subs(u=1,(d iff(g(50),u,u)+diff(g(50),u)-diff(g(50),u)^2)),30);" }}{PARA 11 d"" 1 " " {XPPMATH 20 "6#$\"?![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" }{TeEXT -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$$\"#AF*$f\"3(* ************p7!#@7$$\"#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\"" }{TEgXT -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:" }}h}{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:#\"*iV \"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:=listtojrec(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/kdsol1: 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:=factorl(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,evalfm(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 "" {MPLnTEXT 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#\"\"!FFo/-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#!\"\p"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/q\"\"#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 {rPARA 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 sDEC-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 snatura7natural/ ¿ Í '3( J@àA-B†P¼Q!vnaumÕ\nauto /navg‰\nayiëlnb/àŠ ûŒk˜@ãMÕ\BkëlÔmnba¿ nbb†PnbcfŠ nbcgŠ  nbdpiecesnŠ  nbdpiecesngŠ nbentÕ\nbentgÕ\nbhŠ nbhgŠ nbinIvnbinarBknbpŠ nbsŒW5ëlnbsearch Œ}onbstëlnbstgëlnbter£ nbterm£ nbtre"nbutÕ\nbverticÕ\nbvgŠ recovëlrecover W5$@recreat:yrecto rectodiffeqà¿ K[,†P-[` rectodiffeqg¿ µ%àA†P:y rectohomrec K6 rectohomrecgµ%àAÎj:yrectopràA rectoprocŠ ¿ ÖKÎjël:y rectoprocg¿ µ%àA†PÎj:yrecuç J@:yrecur V1jrecurrãM†P•rrecurrenàA recurrenc³¥àŠ ¿ ûŒÖkK¿Í "µ%3(`+[,‡/è/V167J@àA-BãM*O†P ‚Q¼Q-[_)``ÎjBkëlÔmo}oÊo•ròs!v:yrecurs ¿ 'W5Õ\jBk recursive Š ç ‰\Õ\Ü_jod¬evalbôewfduÄexacttrgœ exampÖ examplì ½8¬ibmintellinuxmaplinputcourimathtimehyperlinkcommoutputhelpheadnormalutputlistitembulletdashfunctgroebnfglmalgogeneralpurposparametrizabliteratioalgorithmbasedusagcallsequencnfprocnfprocgfdprocfdprocgtmproctmprocgtgparametercomputformfunctiofinddependencdecidterminattermorderorealgebrasynopsidescriptfunctionperformiteratlaomputbasiwithrespectskewpolynomialwhosunderzerocommonusesarbasechangfirstknownordesecondanothequationdefinexamplbelowmorespecificaltakeargumentnfnfgtablalreadindicthcorrespondentractualreturnstorintoargumentsmimiggeneratmonoidealsettermsstablproductsametypeaboveithnontriviallinearombinaisonevaluatnfprocgfailfailgnosuchexistuseieselemheprocedurusedstopbordbordergstilldealtmonoidealgtordtordgwitenumeratexpresseqeqgallnormaduringcalculatfoundwarndoestrydetectdimensionalcasemayloopforevalledimplementatifollowarticlefficico!mension¿ ment# Š Æ"”E†Pý\îx:ymention P èmeratÎjmergŠ Õ\òsmericalŠ merit' meromorph†P meromorphic¼Qmes¿ 9' ‰\Õ\messag:met£ metaîxmetalicW5meth†PmethanW5metho £ :ymethod7àŠ ë ¿ £ P ¯"'‡/W5$@J@àA-B”E.F†P!Q¼QÕ\†ijÎjBko sòs!v:y methodologàAmetric àA:ymeuëlmfCxŠ ¿ 9"77Â;J@àA.F‰\Õ\Îjëlòsîx:ymfaÕ\mfb¿ mfc¿ mfcelÕ\mfdëlmfduÕ\mfdufgÕ\mfeÕ\mfesÕ\mfetÎjmffÎjÁq•r!v:yŠ}fgal Õ\ëlfgamÕ\fganfÕ\fgapÕ\fgarÕ\fgarfÕ\fgauÕ\fgblÕ\fgbmÕ\fgbqÕ\fgbsÕ\fgcÕ\fgclbÕ\bfgclfÕ\fgclfajoÕ\fgclfajqÕ\fgclfbÕ\fgclfbimÕ\fgclfbinÕ\fgclfcÕ\fgclfcapÕ\fgclfccmÕ\fgclfcemÕ\fgclfcfpÕ\fgclfchlÕ\fgclfdÕ\fgclfdeoÕ\fgclfdipÕ\fgclfeflÕ\fgclfeioÕ\mb¿ àA†PÕ\ëlmbaëlmbbbbbbbÕ\mberW5”E¼QBkëlIv:ymbfduÕ\mbiÕ\ mbinatorialP /îxmbintr:mbintre :" mbintreeg"mbjëlmbmÕ\mbqÕ\mbstruct' àç ï$@ãM¼QÕ\òs:ymbxÕ\mc àŠ ¿ Õ\ëlmcandiÕ\ mcandidatÕ\mcguyk¿ mchÕ\mchyÕ\mcuÕ\mcxtÎjmdŠ "Õ\Îjëlòs:ymdf"mdfdimÎjmdgòsmdistriîxmdjël¼Q‰\Õ\ ý\å] ^Ü_djÎjBkëlÔmoÊoòs!vîx:y coaccessiblË-coalŠ Æspecgkspeci$@˜]jspecia£ specialS¿ £ P µ%'ˆ;$@J@àA»M*O‚Qý\djÎjBkŸp!v:ypupj:ypuqÕ\puqdÕ\pur¶Jpure àÍ W5J@¶J‚Q:ypurpoÂ;purpos# .7½8†P ^Îjòs:yput3 Š ‚ è"µ%V1<$@¼Qjoputat ^puteCµ%‡/BkputtŠ ¼QÕ\Bkpuv¿ pvŠ ¿ Õ\ël:ypvaÕ\pvdvëlpvefÕ\pvem"pvf Õ\ëlpvgccfÕ\pvgjëlpvlvŠ pvlzejBkpvmaëlpvsuëlpvtÎjpvtkÕ\pvwëlpvxl¿ pwàŠ ¿ "W5Õ\ël pwantÕ\pwbëlpwcpwdÕ\pwevigycëlpwgfpwgfgpwkf £ ¼Qpwr"pwrj¿ pwsd¿ pwuëlpwvc¿ pwxwkëlpx à¿ Õ\Îjël functiong¿  fundamentalŠ funeq o funeqgofunfëlfung:yfungraBkfungrapBkfungraph Bkîxfuqfurt»Mfurth+ 3£ :?‡/R6$@-B*O‰\ furthermor ‡/futurú+ ,-B!QWVd9qfuyŠ fv7Š ¿ 9<J@†P¼QÕ\Îjël t:yfvbÕ\fvblmyjÎjfvdëlfveëlfvf/¿ Í ½4W5½8ˆ;J@†P ^:yfvfa†Pfvfdfvfg¿ fvflëlfvfncëlfvfp¿ fvfq¿ fvmaëlú+ <dÎjBkëlòst:y½8J@˜@.F‚Q‰\˜]Îjëlòs:yëlatomic ‰\atonË-tatorialÆW5Bksaying'sayssb .FëlsbapëlsbfëlsbfdëlsbhàAsbhnŠ sbjÕ\sbmmtf¿ sbmnÕ\sboëlsbtevëlsc#¿ £ W5àAÕ\Îjëlòsscal'Š £ "!v:yscalen£ scalez£  scalezeta £ scalinggŠ ¿ :ysceamrëlsceuÕ\scewcnëlscgÕ\scggÕ\schemÎjjffdmÕ\jfhfl‡/jfhn7jfhnfàAjfjÎjjfjfn‡/jfjn:yjfzëljgGŠ ¿ 39µ%7àA¶J*O¼QªYÕ\Îjël9qòs:y jgca¿ jgf Š ÎjÊo:yjgj¿ jgrkqëljgx¿ jgyëljh ¿ Õ\ëljhja¿ jhoŠ jhuàji ¿ Õ\jif¿ jigëljiqÕ\jix¿ jj Š ¿ Õ\Îjëljjf"jjjÎjjjjjÕ\jjlëljjlll"jjpyw¼QjjrÎjjk †PÕ\ÎjëljkdkÕ\jkhjÕ\jkjfÕ\ëlni¿ †PÕ\ëlnibÎjnice¿ W5¼QÎj!vnicod P •rdesignà¼Q!vfepf ½8Õ\fepfdfepfepŠ  fepfepfhqfhqŠ feq# Š ¿ µ%Õ\ëloòsfeqfëlfeqfjpfeqnsg ::yfer W5Îjël:yferencÊoferentŠ  ferentialÍ ‚Q¼Qferf Ö:yfermatBkfesR"Oël:yfesf "7fesfeglÕ\festoon¿ fet¿ Õ\Îjëlfetfµ% fetfftfgtµ%feu ¿ Õ\:yfdqfb:yfdrÕ\ël:yfdrfàA fdrfaofaof¿ fdrzàAfdsŠ ël:yfdsf:yfdt ël:yfdtf:yfduë¿ ¼QÕ\áëllgdegfg¿ µ%àAÎj:ychberg!vchebyd chebyshev½4J@†P chebyshevtJ@ chebyshevtgJ@ chebyshevudchecko4àŠ ¿ Æ3(‡/W5Þ5½8J@àA-B†P¼Q‰\Õ\ ^Ì^_Ü_dÎjBkël:ycheckedgq8˜]Š} checkgrammarË-checkin¼Q cheesboardŠ chekdchemiW5chemicalàW5!vchemist¼Qches Š ï chessboardŠ chfe†Pchflëlchhamm ,chidchicago¼QchiefàA ¿ Œ:ÖkÍ 'Ô+[,W5$@˜@ A”E»I¶JãM†P¼Q1VÕ\ý\§bjÎjBkÔmo}oŸp s!vIvîx:yŠ}integer?k¿Í '‡/S:¶JãM¼Qý\BkÔm!vîx:y integergf ¿ îxmyfëlmyihÕ\myjël:J@-BÕ\Bkëlòstur †Ptrick Š Õ\trieˆ;$@†iuytriedĽX98@¤N)`uytring.FtriplÎj trisubstitutW5 trivariat Îjëltrivialµ%½8!Q!vtroduct!vtroublÕ\tru AtructKç 9Æk"'3(”E¼QÕ\˜]jÎjBkŸpIv:ytructor '˜@tructurký\BkŸptrue'3¶J1Vžd†iBkŸp:ytruegK3ìÖË-/Þ5q8ÀDCN˜]Ì^_ždëlÁqŠ}truegc¿ ÖàAÎjël:ytruegf_truncaàtruncat¿ 9W5 A¼QÜ_jtruwÕ\try3Š 9k"'½8$@-B†PÎjÔmtrying9†P‰\tsC¿ ç £ ìk¿½8˜@Õ\BkÔmoÊo9q:yòsevalb‡/½8†PBkiveriserecurrenclgndfunctionprsumkgfprgsumgdeterminexplicitimplementfollowingprococaloptionremembelifelseexpandconvertseqfiendinstancthajustoccuronlychoosmiddlgetmomentsuccessdifferentiatsubsdiffxsnvkbdddqdwffhidhebdvculxvdezzekvehydpbriymjovevalfcbzfwplllfxfzwfewfdxrwwvlvhwfdubuwfhtltoswrsuggestmeanlargconstantapproximatoccupancratioeasiestheuristicapproachmomentlinearreasonablholonomictypethuscomputfewdozeninitiallaplacegalgebraicsubsgalgeqtodiffeqgalgeqtoseriesgalgfuntoalgeqgborelgcauchyproductgdiffeqdiffeqgdiffeqtorecgguesseqngguessgfghadamardproductgholexprtodiffeqginvborelglisttoalgeqglisttodiffeqglisttohypergeomglisttolistglisttoratpolyglisttorecglisttoseriesglisttoseregfglgdegfglgdogfgogfgrevegfglisttoserrevogfgmaxdegcoeffgmaxdegeqngmaxordereqngmindegcoeffgmindegeqngminordereqngoptionsgfgpoltodiffeqgpoltorecgratpolytocoeffgrecrecgrectodiffeqgrectopro‚cgseriestoalgeqgseriestodiffeqgseriestohypergeomgseriestolistgseriestoratpolygseriestorecgseriestoseriesgferecurrencetransformintodifferentialrectodiffeqodeopodegzgygdiffgnowsureexistencclosformaveragsincanydeordersolvablquadraturdsolvcommanddoesjobfactorexpgcancheckconsistencknownserinioaohzzhmojjqhtjvvbvaapejkvjclhndjskhwvoiiqzurxqfiqqhftjvrymxnkpogalsoquitprocesmakesenscomparwellhavenusedallguessphasunexpectatmeromorphpoleanalyscoefficientsolutioexaminsingulardoublmapzgfprinciplsingularitanalysienoughpartshowssatisfformulaexpcoeffngfaopqeblkphffactextremegoodboutdooduvvphblkkecmulbeogdxkrngrszduaioparticularundempiricalprecisedistributionalwhenevpossiblealgorithmhowcharactristiccaseshowstandarddeviatdistributsqrtsqrtgmarkovchebyshevinequalitalmostconfiguratmustpredictaccesssecondfactorialeyqcepgvzfficbhchjkesaijdjyhnveafonlmidjupcdccƒqnqelisttoraimsdetectplausibllinearpolynomialusesparametermaxordereqnmaxdegcoeffdefaultrespectivetunequickdiscoversimplnreclisttofailgcontrolsethighmaxdgcoeffgfungmaxorderqngkfgfhffaftfcffrffvffrfbofafvfafunctiondegrerathremarkablsolvdgfwfqfwfdfwffwfinfinfyfvfwfwfkfenfinfinfjnfwfinfinfyfbofenfgnofinfhofgofwfinfinofcogfinpfwfinfinnoterationalentailsreductcomplteexistsucceedbypasinginfolevelpolylinearodeulerndsolvffeqexpsoltryingexponentialsolutionpartialsuccessfulsolvericcatiartexponesultintpartvariancleadformulasingsimplifsinggvarasymptollectbinomialasymptgagainextremasymptcjimiffnbbpassobtaintheoremvarianceqcjjfthiseemnewconvergencfastaccuratnlyaboutdeviationentailconcentratarounddistributionsasymptoticalgaussiandistrsortevalblinalgtransposmatrixmatrixgnnfechfefbstrstylpointcurvesgafbfftdfefnnfkffhfkcolourgrgbgstylegointgeithbadassignmou„tlawprovadaptbivariatpatternbinarsearchtreemartinezgourdonfattmenextendbgsideearliconsidereliesgbgbggeneratfollowgivenizeparametlocalnendarrivalleavoppositavailablotherwistgktponphzhmzegugsuuynmijqckliizrozpcohhitcnpzgyugnvdqyhfygfegfoccupatlikebeforattemptolvemaxdegcoeffrecbrecbgfkfequatihencsolbdsolmethodsbernoulliinearsolstsolbgigerfgpigfxpgfkgausianerrorerfapparsingbsingbgigffjffffjffafaingbpigexpgcorrespondasymptoticmbehdnjxzdsiyyngsvsonceextremelevenrelativesmallfwmvbufiktxqueakryooydvuohorgfofrjiawlastexampldemonstratinterestpreservinconditmanagconsistentespecialusefulsamefinalsituatoccupiedyxjlwwtqbkdboccupationrecbgegcoeffintegralscubicdegreemethodiffeqlinearsololbgintgfcfugffdoccupatiintegralevaluatnulvupproximatstillverywithinalthoughregimlittllongestablisheapiqicrigtzafrszegezszahhpghmaconclusourpurposdemo…ntratnaturalprobabilisticproblemtoolcomebackthinkhistorsummattheorholonomicbasedroughboundequatiosnsufficevalidatrediscovergeneraldavidrothmawholsessincludcomputationcpuoriginaloccuptatbestorkyhlbzmpricpaidaccesabousawrepeatedobtaineddistributionalrepresentativewillobservpracticeffrecbrecbgfkfequatihencsolbdsolmethodsbernoulliinearsolstsolbgigerfgpigfxpgfkgausianerrorerfapparsingbsingbgigffjffffjffafaingbpigexpgcorrespondasymptoticmbehdnjxzdsiyyngsvsonceextremelevenrelativesmallfwmvbufiktxqueakryooydvuohorgfofrjiawlastexampldemonstratinterestpreservinconditmanagconsistentespecialusefulsamefinalsituatoccupiedyxjlwwtqbkdboccupationrecbgegcoeffintegralscubicdegreemethodiffeqlinearsololbgintgfcfugffdoccupatiintegralevaluatnulvupproximatstillverywithinalthoughregimlittllongestablisheapiqicrigtzafrszegezszahhpghmaconclusourpurposdemo…ëlˆ¡{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 ""ˆ 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 marŠk (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\"" Œ}{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$Ž%\"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 "" {M“PLTEXT 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;/-Fš'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$$\"$]$F*$\"?a%ohoj\"Qoc%>y9fI#F\ž\o7$$\"$g$F*$ \"?%*pVGj)Hw@p*y=\"GN#F\\o7$$\"$q$F*$\"?*RKOS(=HB*=\">e#oR#F\\o7$$\"$! QF*$\"?q)*)>Sw,)>O!HCszV#F\\o7$$\"$!RF*$\"?pI6k<(\\?.\\#RVFwCF\\o7$$\" $+%F*$\"?nx!o>[6\"\\h'f4i<^#F\\o7$$\"$5%F*$\"?qcf[by-\"3)Q,;ZWDF\\o7$$ \"$?%F*$\"?^Af)R:&[(=+URWWd#F\\o7$$\"$I%F*$\"?I/BgO0bDgkeos,EF\\o7$$\" $S%F*$\"?.i>7Ka=\\:Wi'pji#F\\o7$$\"$]%F*$\"?pUj7S(fBQs[BF%[EF\\o7$$\"$ g%F*$\"?\"*pI'o4**)4+(>]]T.-&o#F\\o7$ $\"$![F*$\"?REMnrZ7+\"f1Oz'*p#F\\o7$$\"$!\\F*$\"?!\\^\"))=&\\%f,N>$**> r#F\\o7$$\"$+&F*$\"?fn+g]I;g))Hso/AFF\\o7$$\"$5&F*$\"?ub\"=&pEs%3EI-!* )HFF\\o7$$\"$?&F*$\"?8,()H*)R09:2ByfNFF\\o7$$\"$I&F*$\"?'fP`SUnPOJb4S# RFF\\o7$$\"$S&F*$\"?+?d\"HC6!=$)Q\"=()3u#F\\o7$$\"$]&F*$\"?-*\\?v2*[_8 >2(41u#F\\o7$$\"$g&F*$\"?LK;uFw'*RMss$y%QFF\\o7$$\"$q&F*$\"?()e^;^0%eH rtvjXt#F\\o7$$\"$!eF*$\"?SwBd:eF]N)R9O*GFF\\o7$$\"$!fF*$\"?@h#etC;*>A2 f`m@FF\\o7$$\"$+'F*$\"?13W%GA.(zsFG1#Gr#F\\o7$$\"$5'F*$\"?%zY2[9g\"zc2 Y/Z-FF\\o7$$\"$?'F*$\"?baaW4M9uWxeCo!p#F\\o7$$\"$I'F*$\"?%*)eE)H.\\8I) )fL_xEF\\o7$Ÿ$\"$S'F*$\"?E'eC\"3I[\"[6qzeIm#F\\o7$$\"$]'F*$\"?#e&)G')HQ dp#\\*G`tk#F\\o7$$\"$g'F*$\"?Hf&GG&zIL5(f:q/j#F\\o7$$\"$q'F*$\"?=n;\\m R!fN%[#F\\o7$$\"$S(F*$\"?!\\8*zO/+f1`&**p,Y#F\\o7$$\"$](F*$\"?!yRpl j`FN#\\QWJNCF\\o7$$\"$g(F*$\"?TQ*=Sc**G]@WHQ)4CF\\o7$$\"$q(F*$\"?j8pD% =<(psS6\"*y$Q#F\\o7$$\"$!yF*$\"?t8)4<]t/#oFCH@dBF\\o7$$\"$!zF*$\"?\"HY hQHY3(3GgU:IBF\\o7$$\"$+)F*$\"?zM([')GmwP)*y9cEI#F\\o7$$\"$5)F*$\"?FtL 1BYn)[p%G,wuAF\\o7$$\"$?)F*$\"?I/w+?0js\\Qqi]YAF\\o7$$\"$I)F*$\"?O@#)> t\"Q!)Qyb=Lz@#F\\o7$$\"$S)F*$\"?FzTR9UHdU3[!y!*=#F\\o7$$\"$])F*$\"?Xcp h-Ls[:W9m(*f@F\\o7$$\"$g)F*$\"?d@>]hvm#)otXKmI@F\\o7$$\"$q)F*$\"?UIZOg :\"z6'4J4<,@F\\o7$$\"$!))F*$\"?7M9))z0D(QYCJJ:2#F\\o7$$\"$!*)F*$\"?Qh! HBRe'zKQ5ZxT?F\\o7$$\"$+*F*$\"?$o_cc[AGv*)4:I>,#F\\o7$$\"$5*F*$\"?-Oe6 t?EVXY$RD?)>F\\o7$$\"$?*F*$\"?98#f/F\\o7$$\"$I*F*$\"?#)\\i pWEvIgyc,9A>F\\o7$$\"$S*F*$\"?+\"oa\">c\"33K!>\"4A*=F\\o7$$\"$]*F*$\"? a/-O&p=g\"f*[#oJi=F\\o7$$\"$g*F*$\"?KS/o$*p[B5BL^[K=F\\o7$$\"$q*F*$\"? D#e[(>3md-17[t-=F\\o7$$\"$!)*F*$\"?w_CKRx`e96 rAO%f54!>.'!#N7$F:$\">!)HpS]y7\"H_GZO!z$!#M7$F@$\"?>ddY)))exPJ5=2**H\" F_\\m7$FE$\">D$z7KD/]BbPSVwK!#L7$FJ$\">cufr&o'G=8**yw$[oFf\\m7$FO$\"?H zq&\\()=m?bJlj>E\"Ff\\m7$FT$\">c\\\\Fr<\"Rl[a?$\\5'oT D+V7#QLF37$Fhn$\">'*)HCl;D:PHv)oQ(\\F37$F^o$\">ocz?,9(ysUj+w*4(F37$Fco $\">&)pmzppOFQUx(p$y*F37$Fho$\"?L$=JB:5AKfi)e948F37$F]p$\"?vQnn.nu@9TI (*f3E&\\#*4vRP$[B)*yE=#F>7$Fgp$\">0aUPb&p&)zid,$pt#F>7$F\\ q$\">r3\"z@2$QBkT%fXwLF>7$Faq$\">erkZb$H,bN\\@)e5%F>7$Ffq$\">R1Y='=1+= !e@L$H\\F>7$F[r$\">7g_TSg_NJOM:/&eF>7$F`r$\">0+T#)3w)Rz_Qj?soF>7$Fer$ \">W\"*3)eSoEdzz8D(*zF>7$Fjr$\">hUP*QZ7!f'fsRcF#*F>7$F_s$\"?Gp@pM¡-Pb*> 89jk0\"F>7$Fds$\"?d*)>oHc/KEg:C%4?\"F>7$Fis$\"??!Qg(*[$eFtn_3Cc8F>7$F^ t$\"?j.s5)4^vFBxAcB_\"F>7$Fct$\"?!R;vj;\"=!3!\\_#R#*p\"F>7$Fht$\"?q_.Y !Hf2KIkc&z')=F>7$F]u$\">!Q)\\'GX+*=Wx)y)[3#F\\o7$Fbu$\">5XM3sg(\\$33HQ LH#F\\o7$Fgu$\">(3zb$Q#*32+*z0$>^#F\\o7$F\\v$\">BJtgR**y/$HKFTSFF\\o7$ Fav$\">LG!o+K(\\J7nQ*\\yHF\\o7$Ffv$\">3:8)o(o0u4eNueA$F\\o7$F[w$\">=q& [9ue0axVJ>#[$F\\o7$F`w$\">?'fxo`eU\"\\CX&3ZPF\\o7$Few$\">(o:!y^#G!=b8m d,-%F\\o7$Fjw$\">.N?2%H(*oZ'*o_*4I%F\\o7$F_x$\">>BFUEy'o&4UGl\"*e%F\\o 7$Fdx$\">9:!>VBv`V$)e&=U)[F\\o7$Fix$\">V;3b_vfM-E0#p&=&F\\o7$F^y$\">/v JmY:+jgY-6J\\&F\\o7$Fcy$\">;ksR')GsMYb9\"*f!eF\\o7$Fhy$\">ry#z8Xh$yv`_ SQ7'F\\o7$F]z$\">g*=Y\"**4Aa;ZlhhW'F\\o7$Fbz$\">mKP(o*z@j0bD`Cx'F\\o7$ Fgz$\">W0g8yJ%**3:S?@-rF\\o7$F\\[l$\">i*fAI/u_mkxV$\\V(F\\o7$Fa[l$\">& *)\\\\4FMCI:my6qxF\\o7$Ff[l$\">AN.]>+i_Q?\"GE2\")F\\o7$F[\\l$\">2?I.ru L*>(ehteW)F\\o7$F`\\l$\">@KWsbN)H)yi1gay)F\\o7$Fe\\l$\">^$HFD,89?&HaQb 7*F\\o7$Fj\\l$\">#3>\"o,m*o^VIJjl%*F\\o7$F_]l$\">8tH;di\">tdWmF0)*F\\o 7$Fd]l$\"?B/d`n[!H1!¢Qd6S95F\\o7$Fi]l$\"?VmMF#onAX&o\">.3qF\"F \\o7$Fa`l$\"?#e&QC>'zAJDu(4?38F\\o7$Ff`l$\"?CZ3=3)p2N/.)Q\"*Q8F\\o7$F[ al$\"?,\"of3Ez#H?+mc6p8F\\o7$F`al$\"?S(H;(4%GM8S%*yw()R\"F\\o7$Feal$\" ?\\N]^zWJ!*HTV#pyU\"F\\o7$Fjal$\"?f$f2v9h&*pRCcmjX\"F\\o7$F_bl$\"?P7Ud GXQ0.NAQC%[\"F\\o7$Fdbl$\"?CqCKRQa$=!)ejx9^\"F\\o7$Fibl$\"?Zy<_aB*f5SX 9Y!Q:F\\o7$F^cl$\"?O$G?\\^VS;Aq+HRc\"F\\o7$Fccl$\"?72(yoKm)eSR%Q2\"*e \"F\\o7$Fhcl$\"?5,\"\\KIE(ziKARc8;F\\o7$F]dl$\"?SS$*yeWQ_%)eIFGP;F\\o7 $Fbdl$\"??Y6aWm$HrU+O\\-m\"F\\o7$Fgdl$\"?!e>d5Q`2S`tx]Co\"F\\o7$F\\el$ \"?+bi>SIH;y \"F\\o7$Fefl$\"?+QnmYLgb))z%3J!*z\"F\\o7$Fjfl$\"?5([s-V;#['HG$Qi:=F\\o 7$F_gl$\"?S)zU()e$)*y%z7W%RJ=F\\o7$Fdgl$\"?SK$H£Sl2IYv:?Tj%=F\\o7$Figl$ \"?I/eRi%f>t6X[j/'=F\\o7$F^hl$\"?5F$=sb^M(Q&=qhP(=F\\o7$Fchl$\"?5TP)*R M;*QGwEPi)=F\\o7$Fhhl$\"?+V\"f1*e)z_4#fD*y*=F\\o7$F]il$\"?q(3En^l?v3$y 3t3>F\\o7$Fbil$\"?gzGzy0`<_'>Tc(=>F\\o7$Fgil$\"?S-84TP_-!HO>uz#>F\\o7$ F\\jl$\"?!=)f!pDyiRAW1!RO>F\\o7$Fajl$\"?]f2)>UlkPeYj5S%>F\\o7$Ffjl$\"? ]_Mx7(R*[%*oXK%3&>F\\oFjjl-F$6$7`q7$F($\"?+++++v$4ry\"ex@R(*F\\o7$F/$ \"?5n7'oW#Ge$)pr&)G$Q*F\\o7$F5$\"?T^E*G>_TI;$QqOS!*F\\o7$F:$\"?'>)*HAl %4]T^5z(*4()F\\o7$F@$\"?BhEF#*>i%es\"zJm\"R)F\\o7$FE$\"?WDpeVEI>oct:) \\3)F\\o7$FJ$\"?turh#3(eCek\\z]*y(F\\o7$FO$\"?1_3`]_MNrm+F$[](F\\o7$FT $\"?Q2)*>e=$Q!4?8j'pF\\o7$Fhn$\"?UF!y# GaO37u:Cs6nF\\o7$F^o$\"?*=&HAAO?T`'=!fVmkF\\o7$Fco$\"?QPZ=TAJUu]DO6IiF \\o7$Fho$\"?O982!Q*G*4o(zzU-gF\\o7$F]p$\"?\")pt(3+h.3l/LjIy&F\\o7$Fbp$ \"?m$yWa,T\"z&y&ybrrbF\\o7$Fgp$\"?#oV+I\"*y$HtoQ<4o`F\\o7$F\\q$\"?+E55 0Ke(=6G`4><&F\\o7$Faq$\"?%4)*))>Rg*Qr4**p*G)\\F\\o7$Ffq$\"?KW2v3y7fi![ 6#z+[F\\o7$F[r$\"?Lm$fq#*p!*)*yIVU`i%F\\o7$F`r$\"?m:1*>K%H\"=vHt/jX%F \\o7$¤Fer$\"?3^WwGSexPI#oWMH%F\\o7$Fjr$\"?*fF&*\\>['3Zx7l`OTF\\o7$F_s$ \"?TlB5Z7#zoAqqi`)RF\\o7$Fds$\"?dvBfdCVH!pqp8(RQF\\o7$Fis$\"?F')yaGw(e Z.Qd(Q*p$F\\o7$F^t$\"?'p0U=h`(*H$32)*=kNF\\o7$Fct$\"?saxWmA%)e;1wH$RV$ F\\o7$Fht$\"?(\\Tef%p&yOr$4lV3LF\\o7$F]u$\"?wsDfy(\\.\"R![VEv=$F\\o7$F bu$\"?F&o/x[hznO\"Q^.rIF\\o7$Fgu$\"?>@``vZ-1TxI6!)eHF\\o7$F\\v$\"?slq4 dz)Hf!)e#)o1&GF\\o7$Fav$\"?w)*4#[oXI'[pA$)[YFF\\o7$Ffv$\"?M$4C%Gt1tns) >:hk#F\\o7$F[w$\"?[K?0_N7#\\[&4.T\\DF\\o7$F`w$\"?9(zyZ%fx4D!ffRiX#F\\o 7$Few$\"?Kq'f?p2C2fy*QZmBF\\o7$Fjw$\"?3zC4OjH>awv())*zAF\\o7$F_x$\"?T! z#F\\o7$Fdx$\"?*[&eiXr4pFYt]Q;@F\\o7$Fix$\"?Vcwl#z;DWgSpR !R?F\\o7$F^y$\"?0O<9\"GCH#f\"y(4_k>F\\o7$Fcy$\"?))zRa];[No`@cs#*=F\\o7 $Fhy$\"?o\"\\@\"G;KMyM(4aN#=F\\o7$F]z$\"?Y;Q*4,zr90\"pv\"F\\o7$Fbz$ \"?lqsbo'>))4Iv[-Fp\"F\\o7$Fgz$\"?ao`GVy4F\"4g+T3j\"F\\o7$F\\[l$\"?)y# )G:)f+j(HLJS7d\"F\\o7$Fa[l$\"?GQqyyS!f]Mny09F\\o7$F`\\l$\"?5g`\"*4%>j?`2hP QN\"F\\o7$Fe\\l$\"?*eb[V9P>u'H!G¥gVI\"F\\o7$Fj\\l$\"?4K)=Enf%e'pw9\"pc7 F\\o7$F_]l$\"?NTI'*>q4\"G*\\ITw57F\\o7$Fd]l$\"?=Z<8L\\&fSe9c:l;\"F\\o7 $Fi]l$\"?+@\"[*)))4>30+5%)Q7\"F\\o7$F^^l$\"?f+5!pw&=\\zMZ1\"G3\"F\\o7$ Fc^l$\"?J2D$e\">[`#>XEQK/\"F\\o7$Fh^l$\"?%=fP)>_ei>W$47^+\"F\\o7$F]_l$ \"?OQTKo]QIc\")3Gz$o*F>7$Fb_l$\"?FK,^&f\")Q8r<0*))H$*F>7$Fg_l$\"?=`.!> :(QL)*\\c!>*))*)F>7$F\\`l$\"?'p<\"H7hU,U#f95/m)F>7$Fa`l$\"?OeQ?C*[*)f; u\"p!RM)F>7$Ff`l$\"?57'*H!>M/Z([61(*Q!)F>7$F[al$\"?lFt!e.:9XXN]y^u(F>7 $F`al$\"?)=Dyo\")RE?')yJB@Y(F>7$Feal$\"?8<%Q]NL,g(4jET*=(F>7$Fjal$\"?0 JIG+WJ*f)4)[om#pF>7$F_bl$\"?%e*=:+[?&pd!el_tmF>7$Fdbl$\"?BPNFN')GfWY\\ fjHkF>7$Fibl$\"?sYuTv*[\\=aOceY>'F>7$F^cl$\"?\"[!**yFLVa76e'o#ofF>7$Fc cl$\"?@E3mT&Qk[,\\R_,v&F>7$Fhcl$\"?>'\\:![\"G)oug0u+SbF>7$F]dl$\"?Fze< euheP;sBaP`F>7$Fbdl$\"?`-Q;B))*G(3,BmZU^F>7$Fgdl$\"?(>!\\^EoQI#3SuRX& \\F>7$F\\el$\"?h>0uC(zJmmN?rMx%F>7$Fael$\"?$fr7j;A0$3[\"**>!*f%F>7$Ffe l$\"?MI4ei:9e3'4FW4V%F>7$F[fl$\"?Kl!)*4BWW#3QV5,pUF>7$F`fl$\"?Ie-R'=jQ Mz_#e*H6%F>7$Fefl$\"?^v6p7eQ#[krL#oiRF>7$Fjfl$\"?;DS/m?I/;p¦.A'y\"QF>7$ F_gl$\"?O#G[)3t=:&)zkYLyOF>7$Fdgl$\"?%R7]qFCfX(R(H1Ra$F>7$Figl$\"?UhQJ 7$F^hl$\"?uXj\"*=b+/,dq%3'*G$F>7$Fchl$\"?\"*ouO)RqKIY_[ 'QpJF>7$Fhhl$\"?O#QS\"H#)p7$F]il$\"?!)37+s:IOB-QG'>%HF>7$Fb il$\"?`a\"3SS;H3T^!fWMGF>7$Fgil$\"?*HH)3tJ(zi!f%Ge3t#F>7$F\\jl$\"?-.(* >IjA'GxdPc5j#F>7$Fajl$\"?Oy_g>^8oE8E=!\\`#F>7$Ffjl$\"?H)f4f^R0U>$R8EUC F>Fjjl-F$6$7`qFc[m7$F/$\"+=q%\\4*!#A7$F5$\"?VBlV(\\v-P0]oI5P#!#Q7$F:$ \"?Td&z/Ec0w!H:b!f%QFg[m7$F@$\"?GkTq?gi&\\#G\"yc=@#!#O7$FE$\"?[z2/'o=% 4yJ#pJH\"zFabo7$FJ$\"?a^@n12g/!)>0n#=:#F[\\m7$FO$\"?k$yO8-&fF(Qb%GQ#*[ F[\\m7$FT$\"?/0oL%Q:$oJ!pM]X!)*F[\\m7$FY$\"?NVZ1()pfd]L(=B'G\\-w`!R'[F_\\m7$Fco$\" ?0xXCJ*\\:0P'o(\\VV(F_\\m7$Fho$\"?zN6rRwOAkPd*=G4\"Ff\\m7$F]p$\"?9)3om BG$y$yb(RMa:Ff\\m7$Fbp$\"?9MizslNB:Ol%o$\\@Ff\\m7$Fgp$\"?>7d#[t>(yCJF0 i+HFf\\m7$F\\q$\"?o!GL^(4$QFf\\m7$Faq$\"?@)*\\y+VemNXK`L'Ff\\m7$F[r$\"?JrC,$Q!G&Q(3@p0fzFf\\m7$F`r $\"?M5rp=`]R]-Z9\"G4UG37F37$Fjr$\"?oO:;*\\ =%ehZ#)\\gj9F37$F_s$\"?We/Et5\\6\\Ul<>b$pi&3#F 37$Fis$\"?,i5msm&3MRE$4VdCF37$F^t$\"?,**yow#HtB_NAYI(GF37$Fct$\"?-PQt& )*G7aaiOA[L$F37$Fht$\"?nWB+$z/Y&\\XRB+XQF37$F]u$\"?^Y+=$*o2ylq)35dS%F3 7$Fbu$\"?cg#R\"\\nE?n-D$R*=]F37$Fgu$\"?yI+rfe%=v,7dVlo&F37$F\\v$\"?3%)fV>RPWF-T'F37$Fav$\"?bUL:(\\TvJ0#\\\"R:>(F37$Ffv$\"?#H`27.8RF\\e ;k=.)F37$F[w$\"?VN;y!3>q)z#[2?C$*)F37$F`w$\"?_g,uF*)3\\V))*\\_U*)*F37$ Few$\"?l*4qH(GYnz)4@B=4\"F>7$Fjw$\"?E\">Ep.m\\T.NE00?\"F>7$F_x$\"?W;L) *4b\\L*)\\%)H_:8F>7$Fdx$\"?$3AN?&4Y)eyM95pV\"F>7$Fix$\"?vV8#)zh+LmMwFo k:F>7$F^y$\"?pEW&oYsVG>*Q$R))p\"F>7$Fcy$\"?/mkdq'H'fXWM4OR=F>7$Fhy$\"? @(p%\\)=C1p(\\4;@')>F>7$F]z$\"?w&)4IPYdj$oErQ$R@F>7$Fbz$\"?$zI=l*Q*>TP 0Jt')H#F>7$Fgz$\"?svY)o$[y&**\\blITY#F>7$F\\[l$\"?t+'*Gx7*R_D0q5cj#F>7 $Fa[l$\"?c_#HN#poDypa')*H\"GF>7$Ff[l$\"?Y*z4wuDb\"*[jdlh*HF>7$F[\\l$\" ?5Pmb(o7;0$G(**o\\=$F>7$F`\\l$\"?%y*Rxf+pr=euNDzLF>7$Fe\\l$\"?g-kL=L#f t$Gl<&)yNF>7$Fj\\l$\"?AOlTo#)f#yQ)>Xe$y$F>¨7$F_]l$\"?w_l*RK9E$*RF >7$Fd]l$\"?+lSk/cW)\\a-D%o2UF>7$Fi]l$\"?9)z>T?/`q73-UmU%F>7$F^^l$\"?H2 +!o*f_Y?*HS<*\\YF>7$Fc^l$\"?9JH'o?E)o)*y4YGx[F>7$Fh^l$\"?Y![x$)f1Ddasp 5&3^F>7$F]_l$\"?/-w,8XX5&o6HcLM&F>7$Fb_l$\"?\\MB#*fWw-u)pIw:e&F>7$Fg_l $\"?Q'Q%GbM&4Hu2m?H#eF>7$F\\`l$\"?fMz\"H>%)HGd!p\\8ngF>7$Fa`l$\"?OB(QC A8*GasN7'RJ'F>7$Ff`l$\"?o`&QO+nLlbU_QJc'F>7$F[al$\"?'37h$ep4pat8OS9oF> 7$F`al$\"?![!4d'4*)*))4JP;\\nqF>7$Feal$\"?!*=YUuLw(QZ=qO@K(F>7$Fjal$\" ?*\\$G0;%*RS$Rx[s!yvF>7$F_bl$\"?5H,nT]nK?&z#G.NyF>7$Fdbl$\"?F>vh:/Z%44 bE_F4)F>7$Fibl$\"?kD'[YWR;Gs()em4N)F>7$F^cl$\"?D]1kB/7%**yRM8%4')F>7$F ccl$\"?Pqzp/b%H'yhCB$y'))F>7$Fhcl$\"?]p_t/p#*)yHt.mf7*F>7$F]dl$\"?%Q$R $Qy*3:@7W,c$Q*F>7$Fbdl$\"?/zD!fN&>Esx!)QOS'*F>7$Fgdl$\"?bJ0#G`h0(RU^/8 '*)*F>7$F\\el$\"?t!yGKTc$3@$*R<1:5F\\o7$Fael$\"?'eSh=O4hk(R&pe./\"F\\o 7$Ffel$\"?$y'G*yCu&*[zaj![l5F\\o7$F[fl$\"?@<1c%*4g'*4::[S!4\"F\\o7$F`f l$\"?g[N<)HZ\"Q8<[!4^6\"F\\o7$Fefl$\"?JNuz&)p*fu%eKlPD/[Y9!G\"F\\o7$Fhhl$\"?*H)pvMA!zeXB2lBI \"F\\o7$F]il$\"?%*p/5#>*4st9>X>C8F\\o7$Fbil$\"?s'*[.R-K\"HzSWmO\"F\\o7$F\\jl$\"?Xut5'>Wu\\:f=&=(Q\"F\\o7$Fa jl$\"?w\\g.P\")\\?C1gDI29F\\o7$Ffjl$\"?CEykKv=B\">0`epU\"F\\oFjjl-F$6$ 7`qFc[m7$F/$\">+v=Ud)yX(48r=<)))F_\\m7$F5$\">b*p,u:)QC$*H%yc*o'Ff\\m7$ F:$\"?3*e`BeQR!RVl.5`@Ff\\m7$F@$\">Bk(\\Qaq$RElLQ'*)[F37$FE$\">)>qR3B \\]V9;Uno\"*F37$FJ$\"?Cn/ISApoue#edF_\"F37$FO$\"?#p\">9;wb\"*yBW7hDBF3 7$FT$\"?Rzk4f#RLcs5Z?,M$F37$FY$\">)p=9lrer!eQDJrd%F>7$Fhn$\">OST?N%\\u 1%)p%[S/'F>7$F^o$\">`u*zSK\\u/\\X#4_u(F>7$Fco$\">J(eWU&**Q#)=.#oA#o*F> 7$Fho$\"?L+'\\ZNow&\\X)oIa=\"F>7$F]p$\"?&4Z1\\\\5L:nW#Q&eU\"F>7$Fbp$\" ?aG!R@Myx-ze&f,*o\"F>7$Fgp$\"?ih*=*HavT!)yx.Gu>F>7$F\\q$\"?d#e3g6_=(z \"H=s3G#F>7$Faq$\"?_mvLx,9eZP)p)*yg#F>7$Ffq$\"?\"p!*y/PYD1Lh7oV&HF>7$F [r$\">um3ZWD1A&=9l?>LF\\o7$F`r$\">%[$R2c[I%*ztª,:X\"QopD`J\"[xA^%F\\o7$F_s$\">t35_P2@b\"*>N> (Q\\F\\o7$Fds$\">[zS]=gJuP&F\\o7$Fis$\">)o1p&[*\\.2vj%)3FeF\\o7$F ^t$\">:BMD%f;3#Qo(\\V'G'F\\o7$Fct$\">UlcTcbK*))RfYOf\" GTCoBl>He.HsF\\o7$F]u$\">w9;pyw%R'oROl(4xF\\o7$Fbu$\">XP2`ke0Y^4sL^>)F \\o7$Fgu$\">PAzC\"zO^X@2$RRo)F\\o7$F\\v$\">&*H?DaoTXETD7]<*F\\o7$Fav$ \">D%oIf(p\"QqmXw@n'*F\\o7$Ffv$\"?3Hk+z&f4W\"pCe%f,\"F\\o7$F[w$\"?2n( \\(\\j?s^exw1l5F\\o7$F`w$\"?j[[`#*[rf]'ex&)R6\"F\\o7$Few$\"?km%femUF_8F\\o7$F^y$\"?SnQp HnV'oT&*)f3)R\"F\\o7$Fcy$\"?i3ZG?@9K>sN'3JW\"F\\o7$Fhy$\"?bu4nXJOid>.p F([\"F\\o7$F]z$\"?ld@fFkoZ(pF+I0`\"F\\o7$Fbz$\"?B]=WV!Ru`VRf6Gd\"F\\o7 $Fgz$\"?Ey2QF\\o7$Fd]l$\"?:7LDJUYU@7^%)zK>F\\o7$Fi]l$\"?]aj_Ms\"Gzd4X_='>F \\o7$F^^l$\"?l&>*>PsN#oWA\\p&*)>F\\o7$Fc^l$\"?@'[x5@+&HJOA8%f,#F\\o7$F h^l$\"?nwy6KL(yZb&)[i4/#F\\o7$F]_l$\"?H)4$4a'eYyqR?IY1#F\\o7$Fb_l$\"?^ GhE+&*z72&[@Wp3#F\\o7$Fg_l$\"?)[s*=')oZO8Bpm!z5#F\\o7$F\\`l$\"?Q$[U(4T 12G\\()>_F@F\\o7$Fa`l$\"?DgQSbh1H$\\]t'zX@F\\o7$Ff`l$\"?u?p0v`h_aV(\\R F;#F\\o7$F[al$\"?;J[_5TP.6,e2Oy@F\\o7$F`al$\"?\"z'ozT_:coZyFn#>#F\\o7$ Feal$\"?FcG#fvr+aiB[*o0AF\\o7$Fjal$\"?y*o[5pAdW>:lH!=f?+*G$>D#F\\o7$Fccl$\"?m()>#yT]ROlbF\\vD# F\\o7$Fhcl$\"?=me)\\MQ>Y&zI_+iAF\\o7$F]dl$\"?#fKyF)oe%\\@8RtDs`FIvE#F\\o7$Fgdl$\"?#e`!>&4$>Vl@,%['oAF\\o7$F\\el$\"? qiM?FL$>>YY`!4@ b?tlAF\\o7$F[fl$\"?K%pOYUvJ'*GTvdFE#F\\o7$F`fl$\"?/3\"f0$Q3zw\\;t#)eAF \\o7$Fefl$\"?8U4kQXOpx_P\"oRD#F\\o7$Fjfl$\"?UO/h-N;)QC)3y?[AF\\o7$F_gl $\"?0)o-#¬HEhty7iSdTAF\\o7$Fdgl$\"?A4iE0+1&phgr%4MAF\\o7$Figl$\"?$o)e$y #3;D;\"ei(zDAF\\o7$F^hl$\"?M?KueZS-ylN1r;AF\\o7$Fchl$\"?)=\"*)3nEEe%zU `ho?#F\\o7$Fhhl$\"?6[slcrm=1v7!yi>#F\\o7$F]il$\"?a%)[_5*4[,W^i()\\=#F \\o7$Fbil$\"?8OAIq;e=(f;vkbRg@F\\o7$F\\jl$ \"?@]ZCg7$F/$\"?ZiJMoq=zA BBs'=/'F>7$F5$\"?#>[u7$zF@S**epuH#*F>7$F:$\"?t>'3!\\**HbQ^q[l<7F\\o7$F @$\"?^6\"G!p!3buG8e#p*er*RF\\o7$FO$\"?d0WM1$HNPst`f5=#F\\o7$FT$\"?[V3a8IMA,T-,)RP#F \\o7$FY$\"?w$yn%pq'RwC*zm))\\DF\\o7$Fhn$\"?f$[u.&)\\&*Rc$*=l(4FF\\o7$F ^o$\"?Bpblq397aQ6AbaGF\\o7$Fco$\"?'>i\"yp@OVnRxs8&)HF\\o7$Fho$\"?%zfl* HRJ2+]d]O-JF\\o7$F]p$\"?#feHm7Gne:**H$ F\\o7$Fgp$\"?cB%)*>\")****fBvs@%\\Xcc/7:cQnNF\\o7$F[r$\"?s -?`=y-#*4R68T6OF\\o7$F`r$\"?xCt;w,M0­^,]WXZOF\\o7$Fer$\"?agTy;#4,N`2pRg n$F\\o7$Fjr$\"?\"G`j)4ua(HEl+kwp$F\\o7$F_s$\"?OnMUv)fN#*fj+)z7PF\\o7$F ds$\"?-Vd[..$Q58')>()=s$F\\o7$Fis$\"?C+h\\J%3v<&3(>``s$F\\o7$F^t$\"?)= lpLJ)z49\\s[fBPF\\o7$Fct$\"?xm9aSPAL')o^%*)pr$F\\o7$Fht$\"?(e_iwR^*)=K pc$*eq$F\\o7$F]u$\"?Qcesg'GKM[tDW1p$F\\o7$Fbu$\"?@())o.z!)pW8Z#*f:n$F \\o7$Fgu$\"?2^6r%y_J2UbCT*[OF\\o7$F\\v$\"?AW!\\9=Ou/,9/sIi$F\\o7$Fav$ \"?,uv>XL+JXG4,A%f$F\\o7$Ffv$\"?b)oxrr&Qsp!Q,QEc$F\\o7$F[w$\"?1\"3U&4c Azc*G%QcGNF\\o7$F`w$\"?:zmT-]bMGfG>A#\\$F\\o7$Few$\"??I#4\\&eBny]MN#QX $F\\o7$Fjw$\"?#4TDio:#e]R6vc8MF\\o7$F_x$\"?*))4Z3$QL/!oA#4krLF\\o7$Fdx $\"?>-?CaAUb>SQ'>#GLF\\o7$Fix$\"?h=z+4**))yp^2*oMG$F\\o7$F^y$\"?tQ$)[R $HQGz`*QaPKF\\o7$Fcy$\"?/D>5^_=`Z+--f!>$F\\o7$Fhy$\"?H-bkk)H5p(f`VuUJF \\o7$F]z$\"?=#[q2]#z**)38FMT4$F\\o7$Fbz$\"?P\"3Hh2m/P]yrz[/$F\\o7$Fgz$ \"?vq=Y1_17!fHs#4&*HF\\o7$F\\[l$\"?:*Q$edh7$))e'))z([%HF\\o7$Fa[l$\"?D <=?ymMR-BvKL%*GF\\o7$Ff[l$\"?^oyy -rpe!p#F\\o7$Fj\\l$\"?KKG0D)[g$=.q0kREF\\o7$F_]l$\"?X`auG/\"p_@&pV#))e #F\\o7$Fd]l$\"?VeWu#=K!HLO(*o>QDF\\o7$Fi]l$\"?xB?xV#F\\o7$Fc^l$\"?kT#ew1qtKeO#o'zQ#F\\o7$Fh^l$ \"?R#4nI3k*G#F\\o7$Fb_l$\"?>f\"p& 3/'*y^.V@9TAF\\o7$Fg_l$\"?z9J%***)z!p&H`EIJ>#F\\o7$F\\`l$\"?S.()eGE\\+ ih!QMc9#F\\o7$Fa`l$\"?'p$))zd9?9.0**3o)4#F\\o7$Ff`l$\"?VyTT2mu&)=D6OH_ ?F\\o7$F[al$\"?_oZ`Ph@!)*)\\WQ\\1?F\\o7$F`al$\"?&)e[dFo.-L+61Ih>F\\o7$ Feal$\"?x*y&=wzFa!Q'[2t;>F\\o7$Fjal$\"?r8i?;w([urW0*zs=F\\o7$F_bl$\"?9 T5NR!4*4eW6%=&H=F\\o7$Fdbl$\"?WB'HF-bo$)=&3***oy\"F\\o7$Fibl$\"?\"Rx) \\FQ>NdeaH&\\u\"F\\o7$F^cl$\"?7/(3;&p&**QmkQ&o.^CrN5j;F\\o7$Fhcl$\"?m\\/^$HH4iHK]7Ki\"F\\o7$F]dl$\"?M7O86NS!)fy&*e,% e\"F\\o7$Fbdl$\"?c.2]brH$y%Hoi^X:F\\o7$Fgdl$\"?VM8xcnVhAYG]r2:F\\o7$F \\el$\"?42!=Swnn&)4-b71Z\"F\\o7$Fael$\"?S`n\\%fa^WbtC3UV\"F\\o7$Ffel$ \"?5f1h!R%f'y*GV1])R\"F\\o7$F[fl$\"?J6=Z?5L4D%oV([j8F\\o7$F`fl$\"?mqYf -0&=3BXcl\"H8F\\o7$Fef¯l$\"?BYa%)**\\$3w8$f7`&H\"F\\o7$Fjfl$\"?o&[[(*3W $3M=1,ei7F\\o7$F_gl$\"?wS$)*p?i>L$[%42.B\"F\\o7$Fdgl$\"?!fw]QXVA!*ytm1 ()>\"F\\o7$Figl$\"?q`)Hb)3.*y5F\\o7$F]i l$\"?*z5%GkG?_fwJic]5F\\o7$Fbil$\"?)f@\\I(=0=)y#z&fG-\"F\\o7$Fgil$\">' )=@zsM8hk%[\"\\x&**F\\o7$F\\jl$\"?()=LelH;$QCR\"y.$p*F>7$Fajl$\"?CJ&e9 `go(z^#4vVV*F>7$Ffjl$\"?gier+tR,%\\+Vs;=*F>Fjjl" 2 248 242 242 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 98 "Similarly, the following picture indicates the probabilities that the pattern abab occurs at most " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 26 " times instead of exactly " }{XPPEDIT 18 0 "k" "I\"kG6\"" } {TEXT -1 6 " times" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "plots [display](\{seq(plot([seq([10*i,add(P[k](10*i),k=0..j)],i=1..100)]),j= 0..maxnb)\});" }}{PARA 13 "" 1 "" {INLPLOT "6(-%'CURVESG°6$7`q7$$\"#5\" \"!$\"?+++++vVB!**fa.\"****!#I7$$\"#?F*$\"?N$e&p`6?'e@SHvu)**F-7$$\"#I F*$\"?g*4?g)*ziq:UtTL'**F-7$$\"#SF*$\"?p,'Q7g%R0!G5yKw#**F-7$$\"#]F*$ \"?us2Ih+8I.zNnA\"))*F-7$$\"#gF*$\"?k,0%o$3j+%fK;`\\#)*F-7$$\"#qF*$\"? ,)Q\"eNcU<8!3G&ff(*F-7$$\"#!)F*$\"?jd_(obu)3&R!QA*eo*F-7$$\"#!*F*$\"?' 3bUKAJ4r!y\"HTXg*F-7$$\"$+\"F*$\"?E-xGp_#efG3))*>;&*F-7$$\"$5\"F*$\"?, 6Dly_\"zg(40w[@%*F-7$$\"$?\"F*$\"?7@&yG\\WLv]K6))4K*F-7$$\"$I\"F*$\"?M fj'4Tuc=/H!4D:#*F-7$$\"$S\"F*$\"?I7p.5Lg1\"os.$z/\"*F-7$$\"$]\"F*$\"?t bpg-nnYu#>y+,**)F-7$$\"$g\"F*$\"?Igd[o2w0nIl6jr))F-7$$\"$q\"F*$\"?Qg)) *\\sy$H4@mM\")\\()F-7$$\"$!=F*$\"?rC6aO!=*)z)oJ10D')F-7$$\"$!>F*$\"?:5 bLS'el&RKG1s(\\)F-7$$\"$+#F*$\"?)fdq\"eBp:n#*Hx15I>0?MyF-7$$\"$]#F*$\"?xKe_A6[6EQ 82;)p(F-7$$\"$g#F*$\"?f=\"y5wiK8#o&*3ghvF-7$$\"$q#F*$\"?^')R/ggQ`'))3x SZU(F-7$$\"$!GF*$\"?%)3<@D>b4ZdzYy(G(F-7$$\"$!HF*$\"?\\@#*)p+m?H]xUA4: (F-7$$\"$+$F±*$\"?%3%4iV$3ob.j2IV,(F-7$$\"$5$F*$\"?9H%=$R%yND_@pq\"yoF- 7$$\"$?$F*$\"?[sN2yA%\\7]G1&fUnF-7$$\"$I$F*$\"?EskCgvV32M'F-7$$\"$g$F*$ \"?*=y,c/`auLD@`(3iF-7$$\"$q$F*$\"?a8Tfh\"\\8Z4LW`\\C:Y[fF-7$$\"$!RF*$\"?_+*opaV'RpOKuH?eF-7$$\"$+%F*$\"?+!*yJA? ^x/;(GcNp&F-7$$\"$5%F*$\"?I*)[EFPu4m+g_IobF-7$$\"$?%F*$\"?3dy')*R>Xsk= r/YW&F-7$$\"$I%F*$\"?/vbm,nS@ud,'3DK&F-7$$\"$S%F*$\"?yu+j?Ov1_>t[1-_F- 7$$\"$]%F*$\"?#\\!fk,pm)eTN#eJ$3&F-7$$\"$g%F*$\"?(R*pw#\\^`_X4X)Hm\\F- 7$$\"$q%F*$\"?k)Hk<^4)=o-'yW5&[F-7$$\"$![F*$\"?-_joWdGp/Q0AePZF-7$$\"$ !\\F*$\"?HRsu\\I;R\"o*GP$fi%F-7$$\"$+&F*$\"?.I=h^%F-7$$\"$5 &F*$\"?`b)))pv]_uk>1^\"3WF-7$$\"$?&F*$\"?CJ8jTxp,.v\"fV?I%F-7$$\"$I&F* $\"?jQz;pp4P-8fT!y>%F-7$$\"$S&F*$\"?h@x&Q4]8HbO%zV&4%F-7$$\"$]&F*$\"?z :z%e3>1iC8DZ\\*RF-7$$\"$g&F*$\"?Tk;n(\\3X\\,xrJj*QF-7$$\"$q&F*$\"?![\\ 3([u+33-+&)e*z$F-7$$\"$!eF*$\"?h0i(e6()\\t@)eCr/PF-7$$\"$!fF*$\"?<,:Wy XxqXhHjp6OF-7$$\"$+'F*$\"?ZfeN\"*GN%fn²])3`?NF-7$$\"$5'F*$\"?&*[2\\$)>& 3ex\")307V$F-7$$\"$?'F*$\"?B%o/HRj)oj'oB1PM$F-7$$\"$I'F*$\"?:xdr^2@$fD /5?!eKF-7$$\"$S'F*$\"?Us,7o&[BH7#[58uJF-7$$\"$]'F*$\"?6]J8:'=Cbz4-N'4u>z!frI$HF-7 $$\"$!oF*$\"?kRTWE+zK1Ss1>cGF-7$$\"$!pF*$\"?H,b6TwNDN&[p65y#F-7$$\"$+( F*$\"?/%oi#43IA>zUH^2FF-7$$\"$5(F*$\"?[J'*o68H9y%\\,scj#F-7$$\"$?(F*$ \"?#o^Mi04[g\"G.fYlDF-7$$\"$I(F*$\"?sIiO>&H%z:Dn5(o\\#F-7$$\"$S(F*$\"? ;xpD')Qy#GlM]j)HCF-7$$\"$](F*$\"?e=0/D()o`6&4\")=WO#F-7$$\"$g(F*$\"?g% p(Q%G+a^xAD70I#F-7$$\"$q(F*$\"?GgupCsgo_t5)=\"QAF-7$$\"$!yF*$\"?G**>J3 @\"yO!zVK@x@F-7$$\"$!zF*$\"?F+7&pDliN-I8qx6#F-7$$\"$+)F*$\"?\"Q3F-7$$\"$I)F*$\"?*\\-G6\"o?GN]Y-6%*=F-7$$\"$S)F*$\"?8_(ooa3C(eQq]fT=F -7$$\"$])F*$\"?%yhrNWvQv,UadG,*eUQ\"F-7$ $\"$]*F*$\"?()GU[@I$e=ob^iZM\"F-7$$\"$g*F*$\"?VJ+X8NMEHzpTI18F-7$$\"$q *F*$\"?;[45!zIR_0Lug)o7F-7$$\"$!)*F*$\"?>-$y&H*Qp;q*=%4CB\"F-7$$\"$!** F*$\"?'4Q1^c*\\k]'=pFp>\"F-7$$\"%+5F*$\"?4YDm\"o$>#)o$pP$Ri6F--%'COLOU RG6&%$RGBG$F)!\"\"F*F*-F$6$7`q7$F($\"?+++++vVBk&)**********F-7$F/$\"?Q h0OuBl02T**********F-7$F4$\"?&)G!\\)='4\")4g_(********F-7$F9$\"?Z\\=#f d5;Xd=h*******F-7$F>$\"?]-Rv9_]yn2ix******F-7$FC$\"?%>jK9X\")erS=&>*** ***F-7$FH$\"?%fwJjyZ>.bh)z(*****F-7$FM$\"?R$pQ_MJ&H#=ak\\*****F-7$FR$ \"?x:+\"*\\*eN5I9Y)*)****F-7$FW$\"?G9%[2m^z9aMc8)****F-7$Ffn$\"?dr+U#y UbV]be\"o****F-7$F[o$\"?GNpn'Hz!*f#>Fm[****F-7$F`o$\"?w\">*f-@* ***F-7$Feo$\"?pfr'ye)><(\\MdJ))***F-7$Fjo$\"?3:#)zriU!RC`DF$)***F-7$F_ p$\"?]Y**H))p_xNq)prw***F-7$Fdp$\"?c!zM$*fC%3')\\Lr$o***F-7$Fip$\"?n+4 #yL3!RX\\wPz&***F-7$F^q$\"?HN]kCZd$zxl**4X***F-7$Fcq$\"?](fC.h0C)ff\" \\_H***F-7$Fhq$\"?$eS*´3/&[3tdPZ'3\"***F-7$F]r$\"?1=O@vX?V\"H3'e()))**F -7$Fbr$\"?\"R1E(e*3o&GqqMG')**F-7$Fgr$\"?K/@jn.cPw?<7F$)**F-7$F\\s$\"? ta*zNh5$4JH0.!)z**F-7$Fas$\"?2Y]L'Hg5Cm**F-7$F`t$\"?B[%*GDk5>)4Y0S0'**F-7$Fet $\"?F[L))[Pq\\pE$Q#=a**F-7$Fjt$\"?GozkR%Q<[wO))Gr%**F-7$F_u$\"?DUlE\"> uluqB#4MR**F-7$Fdu$\"?2Ta%)y:#Q`FIa!yI**F-7$Fiu$\"?HnF/c)*>o7s01T@**F- 7$F^v$\"?*)p*yG'\\u(*H;<\\>6**F-7$Fcv$\"?*f*Qxk$*4y`x^$)4+**F-7$Fhv$\" ?yu(Go71u!e_jp3)))*F-7$F]w$\"?#)>yiSh5Q*3&)3G^()*F-7$Fbw$\"?sa'\\\"e\\ **4(H$Q/>h)*F-7$Fgw$\"?3;@>BMv/0R%=Wi%)*F-7$F\\x$\"?DK,lQ-%zlJK.h-$)*F -7$Fax$\"?dpP5[o+wR6$H9K\")*F-7$Ffx$\"?6)GDbQ%*ekl?$*y]z*F-7$F[y$\"?Dz !4\"HtP0XkF;$ex*F-7$F`y$\"?R?UX[;*y`!p33Xb(*F-7$Fey$\"?9<.om-:0JO*o;Rt *F-7$Fjy$\"?ulN\\PqTSKx&H67r*F-7$F_z$\"?Ih`ncv1qO`&[=to*F-7$Fdz$\"?*y` XTYBuXe0'RAi'*F-7$Fiz$\"?y#f__$QP!f]MF:fj*F-7$F^[l$\"?0p`C)zCvv>y#=Q3' *F-7$Fc[l$\"?C&='Rv:uZU(G([hz&*F-7$Fh[l$\"?yr\\x))pT91j#\\2'\\&*F-7$F] \\l$\"?Z(fh$>#>yp#3!fa$=&*F-7$Fb\\l$\"?hv7LFOz1!oµm'G&e[*F-7$Fg\\l$\"?b ^cF2!Rx^ktz+@X*F-7$F\\]l$\"?()>AN.]mI[u,')4<%*F-7$Fa]l$\"?R)fP-.K1\\28 @[3Q*F-7$Ff]l$\"?J$e*fUdF&G(RLKNV$*F-7$F[^l$\"?[_5syTm*fZ='>E^*)F-7$Fh`l$\"?RQu-.+83qdFq&=!*)F-7$F]al$ \"?#yDC\\cR`))\\+#\\M^))F-7$Fbal$\"?hLj^a$R=,)o%oX(*z)F-7$Fgal$\"?JTzS mX')>?U(Q!3Z()F-7$F\\bl$\"?3$4>A?A1D3!e5P$p)F-7$Fabl$\"?Q)R;exN![_QL1k Q')F-7$Ffbl$\"?&o8F)R$y(RY@,H\"He)F-7$F[cl$\"?cg)y%Gj9u5#ek&o%)F-7$Fecl$\"?H!yc#e-&*H*)=@`**4%)F-7$Fjcl$\"?]H.Q #*G3uJ>&HJ0N)F-7$F_dl$\"?;[rlh\"H&=#4Rr*>!H)F-7$Fddl$\"?_HRtQl4&HXY'pf%yF-7$Fgfl$\"?%eC<\"4[S#*Hl!GH( zxF-7¶$F\\gl$\"?$ord\"[bt:QvuP)Gr(F-7$Fagl$\"?uYy]EgsebulLYXwF-7$Ffgl$ \"?*pw31'=3U`(pM)\\xvF-7$F[hl$\"?>R-g*)3D`EpG!>!4vF-7$F`hl$\"?T?gjq![. K3+qb+W(F-7$Fehl$\"?+XI&=%)))=)zF#fQ1P(F-7$Fjhl$\"?l0a4Wrsoo\"R%yz+tF- 7$F_il$\"?[(eDivUf))[mYj0B(F-7$Fdil$\"?\"on))\\!HA0%*=8`'*frF-7$Fiil$ \"?'p[^goqT%yNYI.*3(F-7$F^jl$\"?+\"otVHo%4Ih4hz$\"?#f0l)QuOv'e8xn)*** *F-7$FC$\"?:>%zrWY1'pV3Vm****F-7$FH$\"?(*\\gk*\\H1/w%[JH****F-7$FM$\"? JOPOEZKu]w\"o(o)***F-7$FR$\"?Fms>K)>q#R<\"eux***F-7$FW$\"?u$4*p*zMD7C5 Nvk***F-7$Ffn$\"?eG[vIv#=9vmr2Z***F-7$F[o$\"?rb[m#e+=f%fG=\\9ilU*)**F-7$Feo$\"?'y%QM'[wR7(e9,u&)**F-7$Fjo$\"?pZ9w/) 3iF?!e7C\")**F-7$F_p$\"?(pp+K,`T4!))3\\%e(**F-7$Fdp$\"?:ltzVwcGB#>$yYp **F-7$Fip$\"?!)*)HgI+n'*G0<#H?'**F-7$F^q$\"?8)Q(4*yh&QU3v6X`**F-7$Fcq$ \"?'o81<*\\SkzVf\"fO%**F-7$Fhq$\"?#)zy/+fH<9K?BeK**F-7$F]r$\"?037L9e!Q 'QW(z`,#**F-7$Fbr$\"?xuz8=@a**[!p·&4J1**F-7$Fgr$\"?1IFC?\"f;n\"[xb*4*)* F-7$F\\s$\"?!yt5,ftP6h6*R:u)*F-7$Fas$\"?6ZNhoJJ&32C=Pd&)*F-7$Ffs$\"?e* =8)Hs.6(=.&3qN)*F-7$F[t$\"?Al&>PLks+CvV0S\")*F-7$F`t$\"?%=$>l3$)363OHh h!z*F-7$Fet$\"?+8t$)>yi\")f6**G$42+r*F-7$Fdu$\"??]'*[SBtEv.&[(ez'*F-7$Fiu$\"?1OaV;* 4M'>\\K$ptk*F-7$F^v$\"?cT4\"GkZiw\"\\y\\M8'*F-7$Fcv$\"?\"4e#*y[USS%>;4 ^x&*F-7$Fhv$\"?g/-Q&QZoE[\"\\w')R&*F-7$F]w$\"?iB-v.w%Q-kKa>/]*F-7$Fbw$ \"?&y\\pjqm>>%>sY'e,.8&fm!*F-7$F_z$\"?kG;!)f&\\o5$)*fJ25!*F -7$Fdz$\"?XK&4gG![nLacF+_*)F-7$Fiz$\"?;$**HAz*4DfoN=U#*))F-7$F^[l$\"?5 qeHF04bW?T+PJ))F-7$Fc[l$\"?-]e*eb@^Rq;f)))o()F-7$Fh[l$\"?r^>u<&z]TV58? ]q)F-7$F]\\l$\"?Elrjjc$[\"[X$e3)R')F-7$Fb\\l$\"?5#)R![h!Q0GP7!*Ht&)F-7 $Fg\\l$\"?tgWf0C%3+@V[Pb])F-7$F\\]l$\"?uY#*=Y([()4(GP4dO%)F¸-7$Fa]l$\"? ;%*=qirsFu#R0ZkO)F-7$Ff]l$\"?)o6E.13NeW'RT@&H)F-7$F[^l$\"?=j%*R^LTc&*e 05#HA)F-7$F`^l$\"?5.dPRB^*Q;m&oh\\\")F-7$Fe^l$\"?X:!oW8!)3K'>%=^`2)F-7 $Fj^l$\"?QkajN(f8RzUmt,+)F-7$F__l$\"?>()3^w5j'[u+1MT#zF-7$Fd_l$\"?\"3t ^**zLYe.(H@GZyF-7$Fi_l$\"?xj))Q-t+hCnQvmpxF-7$F^`l$\"?B]Z6skLR\")3x(R8 p(F-7$Fc`l$\"?<%HBw8m3%Ga\"3[Bh(F-7$Fh`l$\"?Qdx;U2&)y\\dh8uKvF-7$F]al$ \"?Ugz?b6\">v418oDX(F-7$Fbal$\"?7)H,]([_@]FTk(=P(F-7$Fgal$\"?sZ.!*=MI? B)\\#Qr!H(F-7$F\\bl$\"?r!)[ktwBXzlNs74sF-7$Fabl$\"?9GR\\O>\\k]](*H;FrF -7$Ffbl$\"?Qe`I&)fyLXncn'[/(F-7$F[cl$\"??x&eN\"G55'>nV$GipF-7$F`cl$\"? dhjxE[F-zq(>d%zoF-7$Fecl$\"?>zw+bRA]E'))RJkz'F-7$Fjcl$\"?5*)4fL%)p@Zgk &[Kr'F-7$F_dl$\"?'>+;r^#f0l'QN]*HmF-7$Fddl$\"?sLnndJM;\"\\D`xla'F-7$Fi dl$\"?0EJq$e*))f\\**[*pJY'F-7$F^el$\"?r*\\po_(*oq[B^m(zjF-7$Fcel$\"?$3 D#=B#\\].\"ot^S'H'F-7$Fhel$\"?ho_#3vOVZL$*[\")eF-7$Fagl$\"?M9&yCP=d4qT;A\"*z&F-7$Ffgl$\"?p iH@)RA,hjC'[.<¹dF-7$F[hl$\"?47>QK$*zz(QoKd_j&F-7$F`hl$\"?JzAlIY=J*zBV=Q b&F-7$Fehl$\"?+-R>^H!RXoI.YFZ&F-7$Fjhl$\"?&zJptihm63c'p1#R&F-7$F_il$\" ?)yqKu<7%oOoaq!=J&F-7$Fdil$\"?Tut*Q;*p-/c>6*>B&F-7$Fiil$\"?;0b9HC*yWN> )Hk_^F-7$F^jl$\"?]@HRsG+LY&\\Z&yt]F-7$Fcjl$\"?(f=gu;O%e=J`/W&*\\F-Fgjl -F$6$7`qF`[m7$F/$\"?>pNWM3Y;y#z-6*****F-7$F4$\"?p:0Uo3>))[d')\\K****F- 7$F9$\"?>bWbCvZKZ%yg3y***F-7$F>$\"?GelULP(*[@-L\"y\\***F-7$FC$\"?8A5([ &fH;3AMc\\!***F-7$FH$\"?IXICxD%f;]EdlS)**F-7$FM$\"?9BnU$**F-7$F[o$\"?EeoD]c0nUnuZ$\\\"**F-7$F`o$\"?LmZ\\]p/I<%RH/E*)* F-7$Feo$\"?$y))o3l4#G;uXq>n)*F-7$Fjo$\"?f+3Fbx(3ctb(elQ)*F-7$F_p$\"?7% z')*y^P\">#H8L%p!)*F-7$Fdp$\"?**oa!35#RCN9%zR?x*F-7$Fip$\"?aJ@+>[[*4h( )*>%Rt*F-7$F^q$\"?[JOOrxuinC08m#p*F-7$Fcq$\"?'f0#G!Q*F-7 $Fas$\"?jn%4,:(*zmf(=_*zJ*F-7$Ffs$\"?qATºC\"G(oSO%R+#*HD*F-7$F[t$\"?2Uh YRxW'=S)R>O&=*F-7$F`t$\"?UmiB_Fw@4Uj')>:\"*F-7$Fet$\"?k`\"4ddf_E/PC*fU !*F-7$Fjt$\"?sp843.*))>0&eNmn*)F-7$F_u$\"?qfBlaAalZ>@P\\!*))F-7$Fdu$\" ?$ysT#\\bfrgJaN>6))F-7$Fiu$\"?61M=iI*zJzq5o)H()F-7$F^v$\"?Jd-)omIC1DR@ Bmk)F-7$Fcv$\"?$=:'))3H3jH]\"4l:c)F-7$Fhv$\"?`P/jN5k%4j:(**zu%)F-7$F]w $\"?*\\P:7rKT'*)RnPV'Q)F-7$Fbw$\"?@J+hkKprfhr!) F-7$Ffx$\"?MzyEQs&pWB-YNU#zF-7$F[y$\"?\"o._F0RfvOc`M%GyF-7$F`y$\"?hZAx Tm-rRTeIuJxF-7$Fey$\"?)Q1I'pcUk(HOtbUj(F-7$Fjy$\"?\\=_v5'4&Q=`F^1OvF-7 $F_z$\"?Ty(fj^5%p&RgchsV(F-7$Fdz$\"?>a(G'oDh)H=$[I$zL(F-7$Fiz$\"?i%G7( *=&H1vGu^;QsF-7$F^[l$\"?F6q]EM(*H3*\\3T!QrF-7$Fc[l$\"?PK+$4t^d\"=#[TTw .(F-7$Fh[l$\"?f98A$Rk3ghYDWq$pF-7$F]\\l$\"?hTMxO8O4O/D^KOoF-7$Fb\\l$\" ?\"[ToL;3J(f^epbNnF-7$Fg\\l$\"?u'H8a7wW$\\U!45[j'F-7$F\\]l$\"?n`O()**z %Q5#RdA:MlF-7$Fa]l$\"?,#e[9$HE&G0Gg[OV'F-7$Ff]l$\"?Qi(*zD3p!z'o)ohLL'F -7$F[^l$\"?`n-?9h0u[M8:NLiF-7$F`^l$\"?*o@)HG@,gKDMbnLhF-7$Fe^l$\"?»yQ,N -o+V3k&p)QMgF-7$Fj^l$\"?4mBa\"3,ng3.YVb$fF-7$F__l$\"?oeZCw:$QxB_%)*=Pe F-7$Fd_l$\"?$f+iP\"p:[AZgaPRdF-7$Fi_l$\"?R!QYE>VRlz6bX@k&F-7$F^`l$\"?) **)3r;.F5)Q?/Vba&F-7$Fc`l$\"?Vtjci2D)Q2Te3'\\aF-7$Fh`l$\"?AEHkJmZvQc.1 Qa`F-7$F]al$\"?^#46M\"fv&*G8_`*)f_F-7$Fbal$\"?&=Wy!>JX\"[7*ep=m^F-7$Fg al$\"?%zl^ys!eWhN$[(Gt]F-7$F\\bl$\"?.etzu=`6;=ppA\")\\F-7$Fabl$\"?17h0 BVyTf**)\\L+*[F-7$Ffbl$\"?Q;*4;OUk]V%\\Kt*z%F-7$F[cl$\"?,LmS[)*H=!*pY0 N5ZF-7$F`cl$\"?\"RPa*3WKQD9Az!>i%F-7$Fecl$\"?,8=-5cG)=n!ohUMXF-7$Fjcl$ \"?=jE\"3b6rA$G$RCzW%F-7$F_dl$\"?g=eYELD[U\\y+UiVF-7$Fddl$\"?!z>'[i+:t DLJ\"HzF%F-7$Fidl$\"?Nj'*\\cir&\\^4%fY%>%F-7$F^el$\"?NYiK%)\\C;>3KM/7T F-7$Fcel$\"?SJHEhFqH,Z=JnISF-7$Fhel$\"?Hu&)=Y()\\S@ib^O]RF-7$F]fl$\"?1 =ct#Q/*)*yy#RG6(QF-7$Fbfl$\"??%Hz99&)pKETUqHz$F-7$Fgfl$\"?KAVBw[-c*)** Qw*er$F-7$F\\gl$\"?QIA@I$RJYY8F:*ROF-7$Fagl$\"?70B@n$e1S3\"[u-lNF-7$Ff gl$\"?'e2x.dh\\)>lOsB\"\\$F-7$F[hl$\"?v\"pQOd%Rx4=\"pY&=MF-7$F`hl$\"?V nLcj>#HZ+\")*o&pM$F-7$Fehl$\"?*QlOXzN_$yJ?!okF$F-7$Fjhl$\"?TLW%or^=5k/ Mzq?$F-7$F_il$\"?vr/8¼20$)\\R-.$*yQJF-7$Fdil$\"?TI&\\)4;f\"yl`b&frIF-7$ Fiil$\"?&\\v+*o1@zmm()\\\\0IF-7$F^jl$\"?EP+\"\\I&)>!*H_TI;$QqOS!*F-7$F9$\"?'>)*HAl%4]T^5z (*4()F-7$F>$\"?BhEF#*>i%es\"zJm\"R)F-7$FC$\"?WDpeVEI>oct:)\\3)F-7$FH$ \"?turh#3(eCek\\z]*y(F-7$FM$\"?1_3`]_MNrm+F$[](F-7$FR$\"?Q2)*>e=$Q!4?8j'pF-7$Ffn$\"?UF!y#GaO37u:Cs6nF-7$F[o$\" ?*=&HAAO?T`'=!fVmkF-7$F`o$\"?QPZ=TAJUu]DO6IiF-7$Feo$\"?O982!Q*G*4o(zzU -gF-7$Fjo$\"?\")pt(3+h.3l/LjIy&F-7$F_p$\"?m$yWa,T\"z&y&ybrrbF-7$Fdp$\" ?#oV+I\"*y$HtoQ<4o`F-7$Fip$\"?+E550Ke(=6G`4><&F-7$F^q$\"?%4)*))>Rg*Qr4 **p*G)\\F-7$Fcq$\"?KW2v3y7fi![6#z+[F-7$Fhq$\"?Lm$fq#*p!*)*yIVU`i%F-7$F ]r$\"?m:1*>K%H\"=vHt/jX%F-7$Fbr$\"?3^WwGSexPI#oWMH%F-7$Fgr$\"?*fF&*\\> ['3Zx7l`OTF-7$F\\s$\"?TlB5Z7#zoAqqi`)RF-7$Fas$\"?dvBfdCVH!pqp8(RQF-7$F fs$\"?F')yaGw(eZ.Qd(Q*p$F-7$F[t$\"?'p0U=h`(*H$32)*=kNF-7$F`t$\"?saxWmA %)e;1wH$RV$F-7$Fet$\"?(\\Tef%p&yOr$4lV3LF-7$Fjt$\½"?wsDfy(\\.\"R![VEv=$ F-7$F_u$\"?F&o/x[hznO\"Q^.rIF-7$Fdu$\"?>@``vZ-1TxI6!)eHF-7$Fiu$\"?slq4 dz)Hf!)e#)o1&GF-7$F^v$\"?w)*4#[oXI'[pA$)[YFF-7$Fcv$\"?M$4C%Gt1tns)>:hk #F-7$Fhv$\"?[K?0_N7#\\[&4.T\\DF-7$F]w$\"?9(zyZ%fx4D!ffRiX#F-7$Fbw$\"?K q'f?p2C2fy*QZmBF-7$Fgw$\"?3zC4OjH>awv())*zAF-7$F\\x$\"?T!z#F-7$Fax$\"?*[&eiXr4pFYt]Q;@F-7$Ffx$\"?Vcwl#z;DWgSpR!R?F-7$F[y$\"?0O< 9\"GCH#f\"y(4_k>F-7$F`y$\"?))zRa];[No`@cs#*=F-7$Fey$\"?o\"\\@\"G;KMyM( 4aN#=F-7$Fjy$\"?Y;Q*4,zr90\"pv\"F-7$F_z$\"?lqsbo'>))4Iv[-Fp\"F-7$Fd z$\"?ao`GVy4F\"4g+T3j\"F-7$Fiz$\"?)y#)G:)f+j(HLJS7d\"F-7$F^[l$\"?GQqyy S!f]Mny09 F-7$F]\\l$\"?5g`\"*4%>j?`2hPQN\"F-7$Fb\\l$\"?*eb[V9P>u'H!GgVI\"F-7$Fg \\l$\"?4K)=Enf%e'pw9\"pc7F-7$F\\]l$\"?NTI'*>q4\"G*\\ITw57F-7$Fa]l$\"?= Z<8L\\&fSe9c:l;\"F-7$Ff]l$\"?+@\"[*)))4>30+5%)Q7\"F-7$F[^l$\"?f+5!pw&= \\zMZ1\"G3\"F-7$F`^l$\"?J2D$e\">[`#>XEQK/\"F-7$Fe^l$\"?%=fP)>_ei>W$47^ +\"F-7$Fj^l$\"?OQTKo]QIc\")3Gz$o*!#J7$F__l$\"?FK,^&f\")Q8r<0*))H$*Fd_q 7$Fd_¾l$\"?=`.!>:(QL)*\\c!>*))*)Fd_q7$Fi_l$\"?'p<\"H7hU,U#f95/m)Fd_q7$F ^`l$\"?OeQ?C*[*)f;u\"p!RM)Fd_q7$Fc`l$\"?57'*H!>M/Z([61(*Q!)Fd_q7$Fh`l$ \"?lFt!e.:9XXN]y^u(Fd_q7$F]al$\"?)=Dyo\")RE?')yJB@Y(Fd_q7$Fbal$\"?8<%Q ]NL,g(4jET*=(Fd_q7$Fgal$\"?0JIG+WJ*f)4)[om#pFd_q7$F\\bl$\"?%e*=:+[?&pd !el_tmFd_q7$Fabl$\"?BPNFN')GfWY\\fjHkFd_q7$Ffbl$\"?sYuTv*[\\=aOceY>'Fd _q7$F[cl$\"?\"[!**yFLVa76e'o#ofFd_q7$F`cl$\"?@E3mT&Qk[,\\R_,v&Fd_q7$Fe cl$\"?>'\\:![\"G)oug0u+SbFd_q7$Fjcl$\"?Fze!\\^EoQI#3SuRX&\\Fd_q7$Fidl$\"?h>0uC(zJ mmN?rMx%Fd_q7$F^el$\"?$fr7j;A0$3[\"**>!*f%Fd_q7$Fcel$\"?MI4ei:9e3'4FW4 V%Fd_q7$Fhel$\"?Kl!)*4BWW#3QV5,pUFd_q7$F]fl$\"?Ie-R'=jQMz_#e*H6%Fd_q7$ Fbfl$\"?^v6p7eQ#[krL#oiRFd_q7$Fgfl$\"?;DS/m?I/;p.A'y\"QFd_q7$F\\gl$\"? O#G[)3t=:&)zkYLyOFd_q7$Fagl$\"?%R7]qFCfX(R(H1Ra$Fd_q7$Ffgl$\"?UhQJ%HFd _q7$F_il$\"?`a\"3SS;H3T^!fWMGFd¿_q7$Fdil$\"?*HH)3tJ(zi!f%Ge3t#Fd_q7$Fii l$\"?-.(*>IjA'GxdPc5j#Fd_q7$F^jl$\"?Oy_g>^8oE8E=!\\`#Fd_q7$Fcjl$\"?H)f 4f^R0U>$R8EUCFd_qFgjl-F$6$7`qF`[m7$F/$\"?Qh0O#R*f+)>%**********F-7$F4$ \"?#Qy^Dnf\\Sq*)*********F-7$F9$\"?_vu(>lRJ+[k*********F-7$F>$\"??B*z. r(yfN$R(********F-7$FC$\"?)zJ^3E*>3Cxk)*******F-7$FH$\"?mK)QB$y#R3AW]* ******F-7$FM$\"?w1*)=@'=\\oY#p&)******F-7$FR$\"?9fQ1L1fsN$p]'******F-7 $FW$\"?v?rW?Mb')HphC******F-7$Ffn$\"?BbJDe=2V$Qp>&)*****F-7$F[o$\"?zoc **e@d,-tL!4#f-axt****F-7$Fip$\"?&R.'z3bD\"Q@^xD'****F-7$F^q$\" ?H?e%*)*G!eWU6Oy%****F-7$Fcq$\"?=v%p\"=v)*[&\\S#yG****F-7$Fhq$\"?I=CZ% yL#=)y1`X!****F-7$F]r$\"?<:B`]FCO=7%)=u)***F-7$Fbr$\"?\"eclP)fDocz7jO) ***F-7$Fgr$\"?p>Pi_X9*RKqE2z***F-7$F\\s$\"?JfDJCbUet%HA_t***F-7$Fas$\" ?yKuQs(o#3tG$p(o'***F-7$Ffs$\"?AQe^XT?PG\"\\C**e***F-7$F[t$\"?dkrHOFRd =`A:(\\***F-7$F`t$\"?g'yY^r=XOs#y#))Q***F-7$Fet$\"?srL\"oÀz\\#*\\hmSKE* **F-7$Fjt$\"?vo(z&3#>vajX)f=\"***F-7$F_u$\"?'[$zveox85i:.`*)**F-7$Fdu$ \"?QTDWP+M^&R(yfk()**F-7$Fiu$\"?**e4XaM9g'e,)G^&)**F-7$F^v$\"?K.0&yP?4 0b'3.6$)**F-7$Fcv$\"?Krf3&\\Q3(QV$*pT!)**F-7$Fhv$\"?8\"fOwJws3uU;6u(** F-7$F]w$\"?V@_!*Hqf\"y2NhqS(**F-7$Fbw$\"?pkmWX7u1&G%fFPq**F-7$Fgw$\"?@ NZ)o-]i%3u5ZHm**F-7$F\\x$\"?*QY['*y*G^:oJL\"='**F-7$Fax$\"?l\"H2L%H&[$ =Y2`!p&**F-7$Ffx$\"?\\Au].]>4.q4sa^**F-7$F[y$\"?#>_%zvX\"QVO:c:d%**F-7 $F`y$\"?*p'=^:Y&Q*\\8-pQR**F-7$Fey$\"?'oyHbo7U(GJ]y`K**F-7$Fjy$\"?KkOA ,Xxw+/n^9D**F-7$F_z$\"?4#>Fj%pE6ue;e=<**F-7$Fdz$\"?Y0S$y%>+dM6Eqj3**F- 7$Fiz$\"?&Gb\")H'HxUJ]VjZ**)*F-7$F^[l$\"?J%H)f!\\$4S&*G$p\"o*))*F-7$Fc [l$\"?>lr:]THR\"40VJ#z)*F-7$Fh[l$\"?\\N1`d#y&>*eBR/\"o)*F-7$F]\\l$\"?D (**Q`A))\\)3aZ*zi&)*F-7$Fb\\l$\"?(e\"\\;ffQ!Q'>V!QP%)*F-7$Fg\\l$\"?<0t 6M))*fR[$\\#f/$)*F-7$F\\]l$\"?:v=vNk$[>&y+[U;)*F-7$Fa]l$\"?*[+-2fw/%HL Omh,)*F-7$Ff]l$\"?7j:,jh!ebyaVp:1vs[Nf>`*F-7$Fgal$\"?\"[A8!3X!R&f>O w)[]*F-7$F\\bl$\"?*f5'Q1(*)QX.3MuoZ*F-7$Fabl$\"?J]\"yt\"G[dh$*fe\"zW*F -7$Ffbl$\"?T**>H%GUz'=4g&4!=%*F-7$F[cl$\"?fDH%3PeNnR\"yP:(Q*F-7$F`cl$ \"?tl[7/dVZPc9yMb$*F-7$Fecl$\"?C2.t[H%)3>#\\#>fA$*F-7$Fjcl$\"?)Gsj2(=f &Q0'4t)))G*F-7$F_dl$\"?11uC(p[6%p)>5OUD*F-7$Fddl$\"?o#)f,#p_T\"\\/b8k= #*F-7$Fidl$\"?yh\"Gr.Rby]l-2@=*F-7$F^el$\"?8nA`$>p1*F-7$F]fl$\"?5jF+VzJF[ a!y$oE!*F-7$Fbfl$\"?k4We7+&zpP!z8a&)*)F-7$Fgfl$\"?:)QIsLV(p.ma6]V*)F-7 $F\\gl$\"?:)R(Q?T9t6]\")Gd+*)F-7$Fagl$\"?hw:QM/&=#Hbyqwc))F-7$Ffgl$\"? h)=5'>+7@yz&)\\47))F-7$F[hl$\"?V,htq/cn;oR&olw)F-7$F`hl$\"?K@[(y,+!e30 [.??()F-7$Fehl$\"?*z-5m2\"zpNikO+t')F-7$Fjhl$\"?fve>Oj#3CkIO#*\\i)F-7$ F_il$\"??%[g_*HEx\"G2\"4=w&)F-7$Fdil$\"?Â2&H%f%G:BwS@M%eE&)F-7$Fiil$\"? Th)e@)[hTLFK#=iZ)F-7$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 Ãgrammar 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 Ä"" {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$%%MarkGF%F 2F5/FKF)/F-%%AtomG/F1FN/F4FN/F7FN" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 Å 211 "we obtain the grammar we started with. It is now possible to stud y longer patterns easily. Here are the different states leading to the probability that the pattern abacab occurs twice in a word of length 5000:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "First the grammar is ge nerated" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "G:=gengram([a,b, a,c,a,b]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"GG<-/%\"wG-%&UnionG6 '%(EpsilonG-%%ProdG6$%\"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< *$F(\"\"%FFÆ=F@*$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(\"#;*$F)\"\"&!#Ç;*$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 "" {TÈEXT -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$e%!#^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+?$\"R1/Lv5ZU> 29z5o8>)=/i&QSa#Rs!#]" }}{PARA 11 "" 1 "" {XPÉPMATH 20 "6$\"%+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$\" S)[N*GAL60[Lu7&)\\>k\"f!)o#!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$\"%+!)$\"Sr%H,g.9m0ffd^^BkA)*\\z!R%z#3F!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+&)$\"SM-I%f!>(Hcv%ft#RHq+@pkw6?tq#!#]" }}{PARA 11 " " 1 "" {XPPMÊATH 20 "6$\"%+!*$\"S:48c+o=d#)=j()e#y#)\\o%oU)G=wo#!#]" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+&*$\"SAa(Hqm+*Q`\"RQm8f%eyo2tTud^E !#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"&++\"$\"SCEWCN4uk-#zc*RJ$\\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! zP<%p+!oVs6]J'!#^7$$\"$+'F($\"RS&\\W#y)=]6zb+(>%=C\"4q/*3eHQ*)FD7$$\"$ +(F($\"SeZQ>@xl&>lYktPD5mw(4UO:P$>\"FD7$$\"$+)F($\"S#G^gxJg*Q9Gjs=5eif !RF%zp(p_\"FD7$$\"$+*F($\"Sx%oXA)GoVc!>07J'*>_e! =4-.FFD7$$\"%+7F($\"SxmFPOv\")y)[U9()RË-w-%>;L>[YWJFD7$$\"%+8F($\"SYQ@^ )e[([S[.N'\\@\\w9vFq1\"[1OFD7$$\"%+9F($\"SYQs-^2i#oQD%>GUU%=oaU$)z!y'3 %FD7$$\"%+:F($\"S$*4`)RH#\\Rz#[!Q7Xm]e!3^sZ)>$e%FD7$$\"%+;F($\"R([)Rb8 TC(R\"o'o$4&!#]7$$\"%+@F($\"R4kI7+YW 29z5o8>)=/i&QSa#RsFfp7$$ \"%+@F($\"R&QEY1twJf0#Gj\"=jV8T'y*[o<$z(Ffp7$$\"%+AF($\"R6'oh$eFiL5!)z )3J\\Vi!*=ydt4^$)Ffp7$$\"%+BF($\"Rlx\"=DCbv'oVgafZUf'[qz<[m6*)Ffp7$$\" %+CF($\"Rnrd5A>EBXxln%*=-a[OF_(4ht%*Ffp7$$\"%+DF($\"S*G.lyI5$onRiZhRsv *)o)3XnuN+\"Ffp7$$\"%+EF($\"S5&\\DH?*[QDQtD%z_Mb<>gi'epf5Ffp7$$\"%+FF( $\"S%)*3Y\"*3!o&)yj'>7Cd822f*3Z0i:6Ffp7$$\"%+GF($\"S&H<5*))4\\s!Gy-d(z ^bZ3U-%[^7<\"Ffp7$$\"%+HF($\"SR%>YL@n,n'=5\"=Z_&H&='*>9-)\\E7Ffp7$$\"% +IF($\"S$[&*fl-_#eE#fZSF$)*)3hE?yxv7G\"Ffp7$$\"%+JF($\"S#eP3'f!H.:B`fM k:ZG6!piaj]N8Ffp7$$\"%+KF($\"S>c+$>z.'4o#e:NAKuk'o_;cq6*Q\"Ffp7$$\"%+L F($\"SIX7,!3&e$>z'3\"Qece1<:S2iS?W\"Ffp7$$\"%+MF($\"S'p8pz`rMb_F,Ì?tdSA !4:Hi\\@%\\\"Ffp7$$\"%+NF($\"S-$z#\\*)o87w/ty/y#\\8R$GFAHeX:Ffp7$$\"%+ OF($\"SJ4*Rli?/z6PQFn0co\"*=(QJ?4'f\"Ffp7$$\"%+PF($\"Sg1>kjK%y2[]aO\\* oQ$G@G3G%pX;Ffp7$$\"%+QF($\"SxV'=@()Q<0%GS1VhHB+%=&o-fM%p\"Ffp7$$\"%+R F($\"S.PR3$4d=6.Js9sTf8jl#=fr+Ux$\\l1B\\@3 AO/9U')y\"Ffp7$$\"%+TF($\"Som5X')3*py0@y=>(e[bgmTq&=U$=Ffp7$$\"%+UF($ \"S1(4w'[cOz%px#f^E]dO<^#3i2(y=Ffp7$$\"%+VF($\"S-v/$*f3+e0 fKP3A>Ffp7$$\"%+WF($\"SUqm+sVX692*HOp!4Y!>T@ZWCV'>Ffp7$$\"%+XF($\"S(=A 15$4Z$ziuB/N^SARc9-B5a+#Ffp7$$\"%+YF($\"S0fW>_?aBjDY18!=)fNT,\")QVKX?F fp7$$\"%+ZF($\"SPly#\\/KxK&[#Q6E\\)*=q7%HYE0%3#Ffp7$$\"%+[F($\"So6a6:H :O6eAjvM@u**4b85Ne@@Ffp7$$\"%+\\F($\"S4!e0MVp[+`A_SNX/rM%[@Jw!z:#Ffp7$ $\"%+]F($\"S.ZskEY])4(fA0&HkiCO$*>9#z,$>#Ffp7$$\"%+^F($\"SdDKGtQ/`81+A -&**3j&R2*oP4pA#Ffp7$$\"%+_F($\"SXJ5]Y9z$)4s%fQQ-Wv,..n&*y&fAFfp7$$\"% +`F($\"S?n!H4eTO1tUNSkV>ixc(yma-\"H#Ffp7$$\"%+aF($\"SNMO8sfk#yk]]J@Moe)*HCFfp7$$\"%+fF($ \"S]\"pv:$QuK\"p%y3x![U4N%y=8*eTX#Ffp7$$\"%+gF($\"SNIud*Q`*>#GqT\"oDh$ 3+>AG]VrZ#Ffp7$$\"%+hF($\"SjwB@66^t*f?)z1i;x%*>Zj0'\\*)\\#Ffp7$$\"%+iF ($\"S$z9RTI#G&yZsHFFp-ZBU%\\(>)e>DFfp7$$\"%+jF($\"S&3eQ!e.(*)Q$*H#Ha;7 #[01x#)=r!RDFfp7$$\"%+kF($\"SE-X\\r`;q_o)4E>*\\5!*[r!=N6ub#Ffp7$$\"%+l F($\"S/Z%)yRJ-v6?q(=)z`Etf;3fAiuDFfp7$$\"%+mF($\"S(*R\"fI0`]Q_$HZFB26G Ysk9#=2f#Ffp7$$\"%+nF($\"SB(fyo@fj)p%yE\"fJ_YC)H2g>9dg#Ffp7$$\"%+oF($ \"S]gd)))>6))HatY]d&G8[F&RfzD'>EFfp7$$\"%+pF($\"SmV'[LH?!>VKJy@KI+%p,4 w;pCj#Ffp7$$\"%+qF($\"S<\\npmo)R/Sz*yG/-wrgdOs4EWEFfp7$$\"%+rF($\"SG?> Y3MphuY^wQ!3\"=o\"o>,L=]l#Ffp7$$\"%+sF($\"ST$=.)41)fa!)Q))y15-/(R**=s( eZm#Ffp7$$\"%+tF($\"S(o,hsKi(o4m`LIaAESU!Hl>+Nn#Ffp7$$\"%+uF($\"Sy\\mf 0#euXGg0KTVXua%eAD3E\"o#Ffp7$$\"%+vF($\"S)[N*GAL60[Lu7&)\\>k\"f! )o#Ffp7$$\"%+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%$wabGÐF2F5/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'ÑF( 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\"" }{TEXÓT -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 TaylorÔ 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#,&FUFÕ,!\"&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 Ö"" {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 "" {XPP×MATH 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 "6$-%'CURVESG6$7aq7$\"\"!F(7$$\" $+\"F($\"SYX.5hLk[*z&)eKd_4G4(z.;D)yK#!#`7$$\"$+#F($\"SsRs4l03(>v%ziq$ H*\\7[:rJBF47$$\"$+%F($ \"SIFF3G/kNRuu)pN_T_FLF\\HAq4%F47$$\"$+&F($\"R#oj..Hv'*=N0T!zP<%p+!oVs 6]J'!#^7$$\"$+'F($\"RS&\\W#y)=]6zb+(>%=C\"4q/*3eHQ*)FD7$$\"$+(F($\"SeZ Q>@xl&>lYktPD5mw(4UO:P$>\"FD7$$\"$+)F($\"S#G^gxJg*Q9Gjs=5eif!RF%zp(p_ \"FD7$$\"$+*F($\"Sx%oXA)GoVc!>07J'*>_e!=4-.FFD7$ $\"%+7F($\"SxmFPOv\")y)[U9()R-w-%>;L>[YWJFD7$$\"%+8F($\"SYQ@^)e[([S[.N '\\@\\w9vFq1\"[1OFD7$$\"%+9F($\"SYQs-^2i#oQD%>GUU%=oaU$)z!y'3Ø%FD7$$\"% +:F($\"S$*4`)RH#\\Rz#[!Q7Xm]e!3^sZ)>$e%FD7$$\"%+;F($\"R([)Rb8TC(R\"o'o$4&!#]7$$\"%+@F($\"R4kI7+YW29z5o8>)=/i&QSa#RsFfp7$$\"%+@F($\" R&QEY1twJf0#Gj\"=jV8T'y*[o<$z(Ffp7$$\"%+AF($\"R6'oh$eFiL5!)z)3J\\Vi!*= ydt4^$)Ffp7$$\"%+BF($\"Rlx\"=DCbv'oVgafZUf'[qz<[m6*)Ffp7$$\"%+CF($\"Rn rd5A>EBXxln%*=-a[OF_(4ht%*Ffp7$$\"%+DF($\"S*G.lyI5$onRiZhRsv*)o)3XnuN+ \"Ffp7$$\"%+EF($\"S5&\\DH?*[QDQtD%z_Mb<>gi'epf5Ffp7$$\"%+FF($\"S%)*3Y \"*3!o&)yj'>7Cd822f*3Z0i:6Ffp7$$\"%+GF($\"S&H<5*))4\\s!Gy-d(z^bZ3U-%[^ 7<\"Ffp7$$\"%+HF($\"SR%>YL@n,n'=5\"=Z_&H&='*>9-)\\E7Ffp7$$\"%+IF($\"S$ [&*fl-_#eE#fZSF$)*)3hE?yxv7G\"Ffp7$$\"%+JF($\"S#eP3'f!H.:B`fMk:ZG6!pia j]N8Ffp7$$\"%+KF($\"S>c+$>z.'4o#e:NAKuk'o_;cq6*Q\"Ffp7$$\"%+LF($\"SIX7 ,!3&e$>z'3\"Qece1<:S2iS?W\"Ffp7$$\"%+MF($\"S'p8pz`rMb_F,?tdSA!4:Hi\\@% \\\"Ffp7$$\"%+NF($\"S-$z#\\*)o87w/ty/y#\\8R$GFAHeX:Ffp7$$\"%+OF($\"SJ4 *Rli?/z6PQFn0co\"Ù*=(QJ?4'f\"Ffp7$$\"%+PF($\"Sg1>kjK%y2[]aO\\*oQ$G@G3G% pX;Ffp7$$\"%+QF($\"SxV'=@()Q<0%GS1VhHB+%=&o-fM%p\"Ffp7$$\"%+RF($\"S.PR 3$4d=6.Js9sTf8jl#=fr+Ux$\\l1B\\@3AO/9U')y \"Ffp7$$\"%+TF($\"Som5X')3*py0@y=>(e[bgmTq&=U$=Ffp7$$\"%+UF($\"S1(4w'[ cOz%px#f^E]dO<^#3i2(y=Ffp7$$\"%+VF($\"S-v/$*f3+e0fKP3A>Ffp 7$$\"%+WF($\"SUqm+sVX692*HOp!4Y!>T@ZWCV'>Ffp7$$\"%+XF($\"S(=A15$4Z$ziu B/N^SARc9-B5a+#Ffp7$$\"%+YF($\"S0fW>_?aBjDY18!=)fNT,\")QVKX?Ffp7$$\"%+ ZF($\"SPly#\\/KxK&[#Q6E\\)*=q7%HYE0%3#Ffp7$$\"%+[F($\"So6a6:H:O6eAjvM@ u**4b85Ne@@Ffp7$$\"%+\\F($\"S4!e0MVp[+`A_SNX/rM%[@Jw!z:#Ffp7$$\"%+]F($ \"S.ZskEY])4(fA0&HkiCO$*>9#z,$>#Ffp7$$\"%+^F($\"SdDKGtQ/`81+A-&**3j&R2 *oP4pA#Ffp7$$\"%+_F($\"SXJ5]Y9z$)4s%fQQ-Wv,..n&*y&fAFfp7$$\"%+`F($\"S? n!H4eTO1tUNSkV>ixc(yma-\"H#Ffp7$$\"%+aF($\"SNMO8sfk#yk]]J@Moe)Ú*HCFfp7$$\"%+fF($\"S]\"pv: $QuK\"p%y3x![U4N%y=8*eTX#Ffp7$$\"%+gF($\"SNIud*Q`*>#GqT\"oDh$3+>AG]VrZ #Ffp7$$\"%+hF($\"SjwB@66^t*f?)z1i;x%*>Zj0'\\*)\\#Ffp7$$\"%+iF($\"S$z9R TI#G&yZsHFFp-ZBU%\\(>)e>DFfp7$$\"%+jF($\"S&3eQ!e.(*)Q$*H#Ha;7#[01x#)=r !RDFfp7$$\"%+kF($\"SE-X\\r`;q_o)4E>*\\5!*[r!=N6ub#Ffp7$$\"%+lF($\"S/Z% )yRJ-v6?q(=)z`Etf;3fAiuDFfp7$$\"%+mF($\"S(*R\"fI0`]Q_$HZFB26GYsk9#=2f# Ffp7$$\"%+nF($\"SB(fyo@fj)p%yE\"fJ_YC)H2g>9dg#Ffp7$$\"%+oF($\"S]gd)))> 6))HatY]d&G8[F&RfzD'>EFfp7$$\"%+pF($\"SmV'[LH?!>VKJy@KI+%p,4w;pCj#Ffp7 $$\"%+qF($\"S<\\npmo)R/Sz*yG/-wrgdOs4EWEFfp7$$\"%+rF($\"SG?>Y3MphuY^wQ !3\"=o\"o>,L=]l#Ffp7$$\"%+sF($\"ST$=.)41)fa!)Q))y15-/(R**=s(eZm#Ffp7$$ \"%+tF($\"S(o,hsKi(o4m`LIaAESU!Hl>+Nn#Ffp7$$\"%+uF($\"Sy\\mf0#euXGg0KT VXua%eAD3E\"o#Ffp7$$\"%+vF($\"S)[N*GAL60[Lu7&)\\>k\"f!)o#Ffp7$$ \"%+wF($\"S[&R?iD,Mo8gL'z)=zFx+.r#Ffp7$$\"%+$)F($\"S^K!QrE\" \\grj!>M?/9hRw)eZ$*35FFfp7$$\"%+%)F($\"S=1wv/z>mF.;%yx\"3^>oRwz#*34FFf p7$$\"%+&)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($\"SY%>N>%fl\" \\Z'=x\"4_$R7j:J-))z1$eF.7$ F0$\"S_Fm#4y:\"=W5eldC#HtgpBci+?e#F.7$F6$\"RSvd!>x#QNxo8+'=q\"y\"*3N?6 NJ&fF47$F;$\"Smn3V3&))eP)=!Revlad$314[&)fk5F47$F@$\"S+\"[*))R%R^DT'*ef a)Qa))oCVq*G6m\"F47$FF$\"S/H&*f-N`n56OiwAc(\\X*es.&)4!Q#F47$FK$\"SW5%[ O%HOXbhgL1V8C4>U` 'o&\\5!*FD7$Fco$\"SGW?**\\(=><()RaHG*G@[\\=L!)R:Y5FD7$Fho$\"S\"R[*o<%z v)zP7u0&fKf#)p-y(31+7FD7$F]p$\"SN&oyL$z1h-5-X#\\]1\"H#3[k=(Qi8FD7$Fbp$ \"S%RTak;)3S%3d2G7-gV]$3W#ffF`\"FD7$Fhp$\"S\\yB=ZwRj?c\"o2[&ov_3*[8M93 r\"FD7$F]q$\"S/_oI_J2:&)[M&**f^&[4^(Q//(>'*=FD7$Fbq$\"SmTlOe%f7S9N!=V& 3#)G_f9YYi&)3#FD7$Fgq$\"Svj\"GW)>&H:u>'4#eivswgJfOuvG#FD7$F\\r$\"Sx^A+ Fv&H*y&>j6UjMi$f=i'G0H\\#FD7$Far$\"S+c4$*pVf]+u;f&e_>!3iIdvjB/FFD7$Ffr $\"SFBwAjHfG!en&fTbB7x#*4?3sD@HFD7$F[s$\"SdP@\")Q1\\O6Gav4h_1G&[d'4cmV JFD7$F`s$\"STHpgHM!*z/s$34_B%zsP7F.YJSQFD7$F_t$\"S&eG)\\/,+w# \\NJ%4_dl'>W$e?0T\"3%FD7$Fdt$\"SorFJ>\"HdMl8)eFunW&='p;<4]EVFD7$Fit$\" S\")y)z7![@pW`(z\">`+FP27m$eH`d%FD7$F^u$\"Sj%olf3P5!\\e3Q4![Z0o\"fg6uk F[FD7$Fcu$\"RVE4Q2G#3#**y\"Hs8[t$>e^8C7K3&Ffp7$Fhu$\"R()=y\"fDzvBzBLwM $)Q?4J#Gr)yT`Ffp7$F]v$\"RMaJxd`\"[$*pE_!\\^w)*yR(3>\"[Jg&Ffp7$Fbv$\"Ru %R$Ra'*pP!zcD0**oRlAjni#oq'eFfp7$Fgv$\"R&4M$))GSvB;s@J3\\Wt3S_+'3$z(=[%odlp:\"*f/)F fp7$F_y$\"RE>upu1q&>X>0[LiO:rfb(RlRK)Ffp7$Fdy$\"RuY&*y0+M^t$3+my01OHLI *o1Eg)Ffp7$Fiy$\"R\\y;K/i5%[qYONmgoF\\q(zpb<)))Ffp7$F^z$\"R4&[!y]OD$Gw DcU*e3EOx'*H2e7;*Ffp7$Fcz$\"Rvps)G8f=B'H#z;&4#f+Ub]NV'4W*Ffp7$Fhz$\"R[ hKxN\"*3f(zfI701(zU1(fo(H2s*Ffp7$F][l$\"Sb+MQm^N_>5wx!>9^3z#f'4VT++\"F fp7$Fb[lÞ$\"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$Fàbjl$\"S AhmLNHq8r[d$Q-&e_/]?F(=&zo@Ffp7$Fgjl$\"Ss!Hg*)RJ\\^.w5L?GVYej!=Lv?'=#F fpF[[m" 2 294 214 214 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 7896 0 0 0 0 0 0 }}} {PARA 5 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 15 " Markovian model" }}{PARA 0 "" 0 "" {TEXT -1 444 "By slightly modifying the automaton, it is possible to consider a Markovian model, where in stead of giving the probabilities of occurrence of each letter, one gi ves the probabilities of transition from one letter to the next one. T he new automaton is almost the same as the previous one, except that t he transitions are marked and three more states are added at the begin ing. Here is the procedure generating the new automaton from the patte rn:" }}{EXCHG {PARA 4 "> " 0 "" {MPLTEXT 1 0 858 "gengram2 := proc (pa ttern::list(\{identical(a), identical(b),identical(c), identical(d)\}) )local i, eq, letter, state, j,alph;alph:=[a,b,c,d];for i to nops(patt ern) do for letter in aálph do for j from 0 to i-1 do if [op(1 .. i-j,p attern)] =[op(j+1 .. i-1,pattern), letter] then state[letter] :=cat(w, op(1 .. i-j,pattern));break fi od;if j=i then state[letter] := cat(w,l etter) fi od;eq[i] := cat(w,op(1 .. i-1,pattern)) =Union(Epsilon,seq(P rod(letter,`if`(i>1,cat(Mark,pattern[i-1],letter),cat(Markini,letter)) ,state[letter]), letter = alph))od;subs(cat(w,op(pattern)) =Prod(Ma rk,w),\{Mark = Epsilon,seq(seq(cat(Mark,i,j)=Epsilon,j=alph),i=alph),s eq(eq[i],i = 1 .. nops(pattern)),seq(letter = Atom,letter=alph), \+ seq(cat(w,i)=Union(Epsilon,seq(Prod(j,cat(Mark,i,j),cat(w,j)),j=alph)) ,i=subs(pattern[1]=NULL,alph)),seq(cat(Markini,i)=Epsilon,i=alph)\})en d;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)gengram2G:6#'%(patternG-%%lis tG6#<&-%*identicalG6#%\"aG-F.6#%\"bG-F.6#%\"cG-F.6#%\"dG6(%\"iG%#eqG%' letterG%&stateG%\"jG%%alphG6\"FAC%>8)7&F0F3F6F9?(8$\"\"\"FH-%%nopsG6#9 $%%trueGC$?&8&FDFMC$?(8(\"\"!FH,&FGFH!\"\"FHFM@$/7#-%#opG6$;FH,&FGFHFS FVFL7$-Fen6$;,&FSFHFHFHFUFLFPC$>&8'6#FP-%$catG6â$%\"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 86ã%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 treatäment 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); nåbst:=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^ræFO>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+*$F(\" \"'F-*&F(F6F2F&F&F-F-" }}}{EXCHG {PARA 0 "" ç0 "" {TEXT -1 19 "The coef ficient of " }{XPPEDIT 18 0 "u^k*z^n" "*&)%\"uG%\"kG\"\"\")%\"zG%\"nGF &" }{TEXT -1 84 " in the Taylor expansion of this rational function is the number of words of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 75 " occurrences of the patterns abab and abacab. Here are the first few terms:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "map(expand,series(\",z,10)); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+9%\"zG\"\"\"\"\"!\"\"%\"\"\"\"#; \"\"#\"#k\"\"$,&\"$b#F%%\"uGF%\"\"%,&\"%;5F%F/\"\")\"\"&,&\"%[SF%F/\"# [\"\"',&\"&Gh\"F%F/\"$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 \+ soè 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 méark 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 pattêern[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$@%/FenFfn>&8%6#FY-%$ëcatG6 $%\"wG-F]o6$;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$ìF,F " 0 "" {MPLTEXT 1 0 67 "subs(gfsolve(\"í,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 multiîple 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 multiîël<decalphaunixmaplinputcourimathtimehyperlinkcommoutputnormaheadplotpatternwordbrunosalvversfebruarworksheetapplombstructcombstructgfunsimplcombinatorialmodelproblemcomputationalbiologstuddnasequenccanviewlongtextalphabetfourletterlargfragmenttabulatparticularhugebasegenefewthousandgivenshortinterestdeterminwhethnumboccurrencvirufarawaymostprobablnumberoccurrencveryimprobablmayhavebiologicalfunctnthecombinatorialrationalallsamelengthequiprobablprobabilityoccurkgdependwayoverlapwithitsellibnamnetblagnalgospecificatunivariatgeneratfunctionpatternababworkoverwefirstconcentratspecificattackproblemrelatusingcombstructwritgrammarescribcorrespondautomatonrecognizwordswrittensuchmarknamedpreseverytimcountgiveoccurencunionepsilprodwaunionepsilonwabwabaatomusecheckngunlabellsizewxialsopossiblprovecomputanguaggfsolvgfsolvegfsolvwabgzgwagfwgfwabagfcg4yourÄàP 'ý\yout¿ yov.Fyowëlyp¿ Õ\ÎjBkëlypaÕ\ypc¿ ypdkÕ\ype¿ ‰\_9qypematch ÆÜ_ypergeom7WVÎj ypergeometrico ypergeomgoypesÜ_ypewr¿ ypical3(ypmÕ\ypo Š Õ\ypx¿ ypxgÕ\ypxtÕ\yq¿ "W5Õ\ëlyqaÕ\yqdgÕ\yqest¿ yqfÕ\yqjfëlyqkuëlyqlÕ\yqlefÕ\yqliÕ\yqoÎjyqoeru¿ yqruh¿ yqt¿ yr¿ "Õ\ÎjëlyraëlyrcÕ\yrghëlyrj àAël ^Ì^à^_Ü_)``§bd*djÎj'BkëlÔmo}oÊo ypŸp9qÁq•ròs!v Ivkwîx:y"uy!vprim å]BkprimariP primpart¿ Í ‡/½8J@Êoprin3principlàW5†PÎj:yprintG3àŠ 'Ë-/R6ÀD»MÕ\˜]Bkëlîx:yŠ}priori¿ àA¼QBk:ypro+ÆàA»I†P ^Îjël9q:yprobÕ\!v:yproba‚ Îjël:yprobabµ%probabi‰\probabil Îjël probabiliël probabilisticW5†PBk:y probabilit77ài‚ P {†P¼QÎjëlt!v:yW56ˆ; $@J@àA.F†P !Q‚Q¼Q‰\Õ\Ü_)``ÎjBkëlo9q Áqòs:yugfSKài½k"‡/ˆ;$@J@.F†P‚Q¼Q ªYà^Îjël 9q:ysgnŠ sgtcÎjsgwëlshŠ àAdÎjëlpattern?~3àiŠ ¿ ˜@†P¼QÎjëlWt!v:yziohÕ\ziqëlziqt¿ zirëlziu Õ\ëlziubëlzizrÎjzj¿ W5Õ\ÎjëlzjbdÎjzjccÕ\zjeuzjÕ\zjf¿ ÎjëlzjkÕ\zjkudëlzjmëlzjmmnëlzjpW5zjptihmëlzjskëlzjwëlzjwg¿ zjyŠ zjzÕ\zjzqëlzk ¿ Õ\ëlòszkbpÎjzke¿ zkevhÎj zkiiiiiiifhÕ\zkoëlzkqëlzktuÕ\zkudhÕ\zkvtëlzkwÕ\zkxr¿ zkylyofëlzkylyofdëlzkzëlzl ¿ W5jÎjëlzlaëlzlc¿ zlcad:yzlec¿ zlfÕ\zlhhwëlX9„helpfunctgfunlisttodiffeqfindlineardifferentialequatgeneratseriestodiffeqsatisfiseriusagcallsequenctypelistparameterlistunknownvariabloptionaltypesynopsidescriptprocedurcomputwithpolynomialcoefficientexpressspecifithesmustcorrespondlisttosertypenamcontainmoreelemtriedorderproviddefaultoptionsgfogfegfusedoutputwhossecondfoundfirstcurrimplementatmaximalmaximumdegrecanchangmodifymaxordereqnmaxdegcoeffgivemanytermpossiblsufficientweregivennosolutmeandoessatisfanylessequalexamplnlnlisttodiffeqxggfygdiffgffdgfcfogfgexpsqrtsgzhsqogdiffgseealsoalso_product,Ore_algebra»Ispecification,combstruct˜@spoly,Groebnerå]structures,combstructý\sum_of_sys,Mgfun-Btakayama_algo,Holonomy3(termorder,Groebner¶Jtestorder,Groebner¶Junivpoly,GroebnerÁqvariance,combstruct”EevalfCXàŠ ¿ "W5J@àA†P¼Q‰\Õ\ ÎjBkëlòsevalfgÕ\evalgdevalu¿ evaluat+à¿ "µ%½8$@J@†PÎjëlevbÎjevcyhfÎjevenCàŠ Í "µ%'ü;-B†P¼Q‰\ždjÎjoevent Îj!veventual iBkever?Æè'Ë-R6LD»M*O¼QÜ_†ijëlòsîxevers¶JeverythŠ 'Õ\everytimël everywherëlevfdëlevffÕ\evid"evidenceviou ¿ µ%evkëlevmwÕ\evoÕ\evolut -Bëlevolv àëlevs¿ W5Õ\evwaÕ\evxyëlew'¿ Í ˆ;<.FÕ\Îjël9qewaa¿ ewe ¿ ëlewed£ eweffpëlewf†PewfduÕ\˜@ ALD¼Qý\!vplottreÕ\plouff àŠ W5¼QÕ\Îj:ypls"plspÕ\plug ¿ J@plugg ¿ Õ\pluralÔmplus;Š 9ÆÀD†P‰\Ü_jÎjBkëlòsîx:yplusexprÆ plusexprgÆplusg 9Æplusgf9plusieur•rplutarchàplwkëlply¿ plyoprg‡/pm$Š ¿ "W5Õ\ÎjëlpmafÕ\pmaksgf"pmbëlpmdÕ\pmf "Õ\pmg Š Õ\pmiÕ\pmkÕ\pmm "ÎjpmmgŠ pmmgfŠ pmmm"pmmmj"pmmmmmÕ\pmmmmmmÕ\pmmmwuh"pmmtÎjpmnpbŠ pmo Õ\ëlecialS¿ £ P µ%'ˆ;$@J@àA»M*O‚Qý\djÎjBkŸp!v:ynby'nc˜@Õ\Ü_Îjncat AnceÍ ƒ4˜@nceg‚ Õ\Îjncel!Qncelldncesonchaiòsnchainòsnclear†PnclusŠ ncmW5ncoeffÂ;ncollect3( ncombinat' ncomposit 'ý\nconsid9ncos|cncount' AÎjncreas¶Jnction' à9:R6‚QëlÊoîx:ynctrnctre òsmmÕ\ ummmmmmtlfÕ\ummmwanl¿ umsÕ\umtf¿ umuëlumwfÕ\un?Š ¿ Œ"†PWV‰\Õ\dÎjëlo•r:yrckDreÁrecordu recursiveþtric€ts_îxassl ¿ ëlssociatç 9Õ\ssrsël ssssssscfduÕ\sstijŠ sstijgŠ stkN3£ P ì:" '3(V1½4W5»M†P¼QÕ\ý\Ì^ÎjBkël9qòs!vkw:ysta࣠àA.F:ystabl ½8dstack Š ¼Qstae:ystag ࣠3(ÎjBk:ystaircas ¿ ¼Q!vstancW5standŠ Õ\òsstandaàAstandard3à"‡/W5R6q8àA»M*O†P standardgàAstanleàstantïstari‚ starn‚  starnormalfor‚ starnormalform ‚ ½starr‚ startW.3àiŠ ¿ £ Í w"W5LD¼Q†ijÎjBkëlot!v:yptoticalBknat¿ nateR6natom˜@natorial à}onatu ¿ 3(natur †Pstructu‡/structur—Óà¿ ç P 9Œ:k¿"'"‡/V1½4W5˜@ A”EãM¼Q‰\Õ\ý\Ì^Ü_§b jÎjBkÔm}oŸp •r!vIv îx structuralW5struvehdstruveldstryW5sttoser¿ †P:ystud#¿ P W5J@!QÕ\ël!vstudiP Õ\Îjstyl ¿ †Pstyleg¿ "†Psu;ài.¿ £ 3(W5J@àA†P‰\Îjël:ysual¼Qsub7àŠ ç 9èk'W5”E‰\ÎjBkòs subcomponentBksubdividè subexpress‚  subfactorial:ysubideal!Qsubject ¿ Îj½Ë-q8ÀD˜]tŠ} autocarprod3½Š}autoco3autocom½autocomb P !v¤N*O†P!Q‚Q¼Q  SWVªYXZ‰\ Õ\ å] ^Ì^à^Ü_dždj Îj Bkël Ômo9qÁq sòs!vkwîx :yvqblovqfW5vqgŠ vqnnëlvqskmfÕ\vr¿ "Õ\ëlvrc¿ vrfëlvrkÕ\vrlzëlvrpëlvrymxnkp†Pvryrrzl¿ vrzëlvs¿ "†PÕ\ ^ÎjëlvsatfÕ\vsekBkvsknxÎjvsont¿ vssssÕ\vszëlvt Îjëlvtjciëlvtlu" vtydvojwofÕ\al#ë ¯7-B‚QÕ\Îj:yfgz:"-Õ\Îjëlfgzf"fgzpfÕ\fhK’à¿ £ " ú+W5J@àA.F¼QÕ\PdÎjBkëlòs:yfhalÕ\dges Õ\Õ\ÎjëlnnamÕ\nnbcŠ nnbvŠ nnc¿ nnectÕ\nnfe†PÕ\ffauÕ\ffavÕ\ffb¿ ffblëlŠ è decalphaunixmaplinputcourimathtimehyperlinkcommoutputourihelpheadnormalbulletitemfunctgroebngsolvpreprocesalgebraicsystemsolvusagcallsequencggxgnznzgparametersetlistgeneratorcommutatidealistindeterminatoptionalnonvanishpolynomialsynopsidescriptcommandpreprocesssystemviewcomputcollectreduclexicographicbasecorrespondsubvarietinitialvarietiesfirstsubdividfactorizateachsubsystempassvariantbuchbergalgorithmfactorallintermediatresultordervariablpreservmuchpossiblhendetermintermusedsubproblemusingnsteadmayintroduclotdeterminismprevcertainquantitconsiderrootomittmeantformgbgbgtgdenotlexicographicbasiubmentionconstraintpolynomialsnotesincoutputtriangularwellsuititerativemphasiputneitheranyparasiticsolutnorlosinghowevermultiplicittakenintoaccountpartpackagcanonlyafterperformwithalwayaccesslongexamplngngsgsgzgygsfplexgkffdmapconvertgsrootofgfcfose+àŒdecalphaunixmaplinputcourimathtimehyperlinkcommoutputnormalheadingheadplotdashitemproblemstatisticalclasificattheorphilippflajoletversjanuardiscusshereatoriginwholcombstructcombstructpackagoctobbernardvancutsemstatisticianuniversitgrenoblwrotusclassificatheorwemakeusehierarchicaltreeneedgeneratrandomsuchaccorduniformlawelementclassifimaytakendistinguishtegersaydoyouknowalgorithmdoingledcooperatinvolvpaulzimmermanbernardcutsemoutgeneralevolveticalcomputsciencvolppfirstimplementatdesignundernamegaiatechnicalnewslettnvanoriginalquestmotivatfollowprogrammstatisticbuildusualproceedsuccessaggregatclosestneighbouramongstexistclasshowcanmeasurwaycarrusefulinformatjustnoiscertaingoodexhibitcharacteristicdepartsignificantoneshencsimulatanalysparametertatisticspecificatstartloadingwithallstructsgcountgdrawgfinishedggfeqnsggfseriesggfsolvegiterstructnextstrucfdfdfJ@fdfefëlfdfenfµ%fdfiëlfdfmÕ\fdfmcëlfdfrfhëlfdfsÕ\fdgiufÕ\fdglëlfdgmÕ\ fdgofgzffz"fdgsÕ\fdgtfÕ\fdgvÕ\fdh W5ëlfdhlëlfdhsÕ\fdiÎjfdilëlfdim Õ\ÎjfdinÕ\fdioÕ\fdiqÕ\fdirÕ\fdis"fdjÕ\fdjlÕ\fdjlfÕ\fdjmÕ\fdjpŽÕ\Žfdjpf"Õ\"fdjpfaÕ\fdjpfahrÕ\fdjpfbÕ\fdjpfbbuÕ\fdjpfbhqÕ\fbvsëlfbw ¿ "Õ\Îjëlfbx ¿ Õ\ëlfby ¿ Õ\fbz Îjëlfbzf:yfbzfjp:y fbzfkffyf:yfcW§Š ¿ ½è"7 J@àA.F¼QÕ\+ÎjBkël oypòst:yfcaÕ\fcalÕ\fcalfÕ\fcalfeclÕ\fcamÕ\fcamfaÕ\fcanÕ\fcbnÕ\fcboÕ\fcbpÕ\fccl Õ\ëlfccqÕ\fccrÕ\fccsÕ\fccuÕ\fcdfW5fcdl Õ\ëlfcdmÕ\fcdmfÕ\fcdnÕ\fcdoÕ\fcdpÕ\fcdqÕ\fcdrfedtÕ\fceëlfcel Õ\ëlfcemÕ\fcf_;Š ¿ £ 3(‡/½47X9ˆ;Â;J@†P¼QÕ\ ^dÎjBkëlyp:yà:'W5 A¼QÜ_Îj:yStirling_numbers,autocomb£ TermOrder,typeÌ^Weyl_algebra,Ore_algebraü;`sys*sys`,Mgfun`sys+sys`,Mgfunagfeqns,combstructç agfmomentsolve,combstruct$@agfseries,combstructç alcohols,autocombW5algebraicsubs,gfunæalgeq_to_dfinite,Holonomyú+algeqtodiffeq,gfun5algeqtoseries,gfun1Valgfuntoalgeq,gfunÐ0algorithm_analysis,combstructjallstructs,combstruct§bannihilators,Ore_algebraˆ;expcount,autoerrorregexpcount,automarkovÀDregexpcount,autopruneË-regexpcount,gramerrorregexpcount,grammarkovÀDtype,ClosedIdealÞ5type,CommAlgebra_type,OreAlgebra_type,ShortTermOrderÌ^type,SkewAlgebra_type,SkewParamAlgebra_type,SkewPolynomial_type,TermOrderÌ^weylü;autocomb,Pollard_algoBkautocomb,Stirling_numbers£ autocomb,alcoholsW5autocomb,balls_and_urns¼Qautocomb,channel_allocation†Pautocomb,hierarchy_treesàautocomb,noncrossing_1autocomb,noncrossing_2Õ\autocomb,permutations:yautocomb,polymer_tilingsŠ autocomb,random_graphsÎjautocomb,self_avoiding_walks¿ combstruct Acombstruct,agfeqnsç combstruct,agfmomentsolve$@combstruct,agfseriesç combstruct,algorithm_analysisjcombstruct,allstructs§bcombstruct,attributes‰\combstruct,countIvcombstruct,drawIv