{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 "courier" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Ou tput" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3 " 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 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 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 256 44 "Combinatorics of Non-Crossing Configurations" }}{PARA 257 "" 0 "" {TEXT -1 22 "F. Cazals, August 1997" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Generalities" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 27 "Non-Crossing configurations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Take " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 0 "" } {TEXT -1 178 " points equally spaced on the unit circle, and draw chor ds between these points with the constraint that no two chords cross o ne-another. The resulting configuration is called a " }{TEXT 257 27 "n on-crossing configuration " }{TEXT -1 611 "and the study of such entit ies originates in the work of Euler and Segner in 1753 for counting tr iangulations of a n-gon. Since then several types of such configuratio ns have been defined, and for example the presence/absence of cycles \+ and the number of connected components in a given configuration define the classes of trees and forests, connected graphs and general graphs . In addition to be of combinatorial interest per se, these configura tions are also important for algorithmic problems arising in computer \+ graphics or computational geometry where they provide simple models fo r real-world situations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 446 "If historically the study of these configurations has been carried out one at a time, it turns out that they all fit in the model of algebraic and analytic combinatorics and are thus amenab le to a unified treatment. More precisely, they can be defined in ter ms of grammars, from which the generating functions can automatically \+ be obtained and used to asymptotically analyse the number of configura tions, the number of connected components, etc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 339 "The goal of this workshe et is to present this chain, from the grammars specification using com bstruct, to the asymptotics of algebraic functions using some features of gfun, together with some plots of random configurations and of the singularities determining the asymptotic behaviour. The reader is ref erred to [FlaNo97] for the details." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 525 "The worksheet is organized as follows. \+ In the rest of this section we define a few functions of null combina torial interest but that we shall need to plot random configurations. \+ In section two we present the grammar specifications for 6 of the main non-crossing configurations together with some plots of random config urations. And section three is devoted to the asymptotic machinery use d to count the number of configurations of a given size, as well as a \+ comparison between the asymptotic estimates and the exact values. " }} {PARA 0 "" 0 "" {TEXT -1 75 "\nBut to begin with, we first load the co mbstruct, gfun and plots libraries:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "with(combstruct):with(gfun):with(plots):" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Appendix" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 154 "Again, we define here a few functions we shall need to p lot NC configurations. The first one returns the number of atoms in a \+ structure, that is its size:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "size:=proc(t) convert(map(size, t), `+`) end:size(Epsilon):=0: s ize(Z):=1:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 203 "Since a configurat ion is defined by edges drawn between points equally spaced on the uni t circle, we first show how to retrieve these pairs of indices from th e grammars to be defined in the next section:" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 33 "Collecting the edges of a NC-tree" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "Since a tree T satisfies T=Product (Z, Sequence( Butterflies)), to plot it we just have to collect its edges which are \+ pairs of indices in 0..n-1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 307 "plotTree:= proc(aTree) local r, nbVertices;\n #--op returns the S equence; 0 is the index of the root on the circle;\n #--1 indicates th at the butterflies attached to the root are counted as a \n #--right w ing \n r:=getTreeEdges(op(2, aTree), 0, 1);\n nbVertices:=size(aTree); \n\n plotSetOfEdges(r, nbVertices)\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 270 "The parameters of getTreeEdges are the following: aSeqO fBtf is a sequence of butterflies; rootIdx is the index of the vertex this sequence is attached to; leftRight (-1 or +1) indicates if this sequence is a left/right wing of the bug. The algorithm works as foll ows:" }}{PARA 0 "" 0 "" {TEXT -1 70 "1.We first compute the indices of the apex vertices of the butterflies" }}{PARA 0 "" 0 "" {TEXT -1 95 " 2.For a given butterfly, we recurse on the 2 wings and attach its apex to the rootIdx parameter" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 985 "getTreeEdges:=proc(aSeqOfBtf, rootIdx, leftRight) \n local i, a, \+ eL, currentBtf, cumul;\n \n if aSeqOfBtf = Epsilon then \{\}\n else\n \+ cumul := rootIdx;\n if leftRight=1 then #--we are processing a right wing\n for i to nops(aSeqOfBtf) do \n currentBtf := op(i, aSeqOf Btf);\n a[i] := cumul + size(op(1, currentBtf)) + 1;\n cumul := cumul + size(currentBtf)\n od; \n else #--and here \+ a left one \n for i from nops(aSeqOfBtf) by -1 to 1 do\n \+ currentBtf := op(i, aSeqOfBtf);\n a[i] := cumul - size(op(3, cur rentBtf)) - 1;\n cumul := cumul - size(currentBtf);\n od;\n fi; \n \n #-- recurses for each Prod(?, Z, ?) in the sequence, wi th ? = E or Seq()\n eL := \{\};\n for i from 1 to nops(aSeqOfBtf) do \n currentBtf := op(i, aSeqOfBtf);\n eL := eL union getTreeEdge s(op(1, currentBtf), a[i], -1);\n eL := eL union \{[a[i], rootIdx] \};\n eL := eL union getTreeEdges(op(3, currentBtf), a[i], 1);\n o d;\n \n #--returns the result\n eL;\n fi;\nend:" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 34 "Collecting the edges of a NC-graph" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 199 "This first procedure recursively collect s the edges of an EA, and is a straightforward application of the EA d efinition above. The parameter ori stands for the index of the lefmost point of the arch:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 595 "get ArchEdges:=proc(arch, ori) local i, offset, res;\n\nif (arch=Epsilon) \+ then res:=\{\}\nelse\n #--we first add the `roof' of the arch\n res := \{[ori, ori+size(arch)]\};\n \n #-----Prod(Z, Seq1, Seq2)\n if (op(1, arch)=Z) then \n res := res union getArchEdges(op(2,ar ch), ori);\n res := res union getArchEdges(op(3,arch), ori+1+size(op( 2,arch)))\n else #--Sequence of arches\n offset:=0;\n #--let's proce ss all the arches in this sequence \n for i from 1 to nops(arch) do\n res := res union getArchEdges(op(i, arch), ori+offset);\n offset \+ := offset + size(op(i,arch))\n od\n fi\nfi;\nres;\nend:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Same thing but to a sequence of arches:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 309 "getArchesSeqEdges:=proc( archesSeq, ori) \n local i, arch, offset, res;\n res:=\{\}; offset:=0; \n \n for i from 1 to nops(archesSeq) do\n arch:= op(i, archesSeq);\n res:= res union getArchEdges(arch, ori+offset);\n offset := offset+ size(arch)\n od;\n #--returns the setOfEdges and the new origin\n res ,ori+offset;\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "And now the main procedure which collects the edges of a Non-Crossing graph:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1102 "getNCGraphEdges:=proc(aNCG raph) \n local res, ori, edges, \n i, j,\n seqOfSeqOrProd, seqOrProd;\n \n #--first, the right ear which is a Seq(EA)\n ori:=1; \n res:=getArchesSeqEdges(op(3,aNCGraph), ori);\n edges:=res[1]; ori : = res[2];\n\n edges := edges union \{[0, ori]\};\n \n #--then the EAs inbetween two successive childs of v_1\n seqOfSeqOrProd := op(5,aNCGr aph);\n \n for i from 1 to nops(seqOfSeqOrProd) do\n seqOrProd:= op(i, seqOfSeqOrProd);\n #--2 connected graphs\n if (op(1, seqOrPr od)=Z) then\n res:=getArchesSeqEdges(op(2, seqOrProd), ori);\n e dges:= edges union res[1]; ori := res[2];\n \n res:=getArchesSeqE dges(op(3, seqOrProd), 1+ori);\n edges:= edges union res[1]; ori := res[2]\n\n #--a Sequence\n else \n res :=getArchesSeqEdges(seqOrProd, ori);\n edges:= edges union res[1]; \+ ori := res[2]\n fi;\n\n #--we need to add the current child to the graph root\n edges := edges union \{[0, ori]\};\n od;\n\n #--and th e left ear\n res:=getArchesSeqEdges(op(4,aNCGraph), ori);\n edges:= ed ges union res[1]; ori := res[2];\n\n #--returns the result\n edges\nen d:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "To plot a NC Graph, we just collect the edges and pass them to the plotSetOfEdges procedure:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "plotNCGraph:=proc(aNCGraph); \n plotSetOfEdges(getNCGraphEdges(aNCGraph), size(aNCGraph))\nend:" }} }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 228 "Now, given pairs of indices in \+ 0,...,nbVertices-1 on the unit circle, the following procedure draws t he corresponding chords\nassuming that the k-th point has coordinates \+ ((cos((2 Pi k)/nbVertices), sin((2 Pi k)/nbVertices)) ):\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 208 "plotSetOfEdges:=proc(aSetOf Edges, nbVertices) local pointsOnCircle;\n\n pointsOnCircle:= expand(m ap(`*`, aSetOfEdges,2*Pi/nbVertices));\n plot([op(map2(map,[cos,sin], \+ pointsOnCircle))],color=blue,axes=NONE)\nend:" }}}}}{SECT 0 {PARA 3 " " 0 "" {TEXT -1 48 "Counting and drawing non-crossing configurations" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 17 "Trees and forests" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 5 "Trees" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 470 "A Tree is a sequence of butterfli es attached to a root, a Butterfly being an ordered pair of trees whos e roots have been merged into a single node. Since this merge step ca nnot be specified by an operation such as B = Prod(T, Z, T)/Z in combs truct, it is more convenient to express a butterfly as the product of \+ 2 forests, a Forest being a sequence of butterflies. Indeed, we now j ust have to attach the roots of all the trees of the two forests to a \+ newly added node:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "tbf:= \{T=Prod(Z, Sequence(B)), F=Sequence(B), B=Prod(F,Z,F)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tbfG<%/%\"TG-%%ProdG6$%\"ZG-%)SequenceG6#%\" BG/%\"FGF,/F/-F)6%F1F+F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Stand ard functionalities of Combstruct consist in counting the number of en tities of a given type:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " seq(count([T,tbf], size=i),i=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6,\"\"\"F#\"\"$\"#7\"#b\"$t#\"%G9\"%_x\"&jK%\"'vmC" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "seq(count([F,tbf], size=i),i=1..10);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"\"\"\"$\"#7\"#b\"$t#\"%G9\"%_x\"&j K%\"'vmC\"(:2V\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "seq(cou nt([F,\{F=Sequence(B), B=Prod(F,Z,F)\}],size=i),i=0..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-\"\"\"F#\"\"$\"#7\"#b\"$t#\"%G9\"%_x\"&jK%\"'vm C\"(:2V\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "And randomly generat ing all the configurations of a given size:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "allstructs([T,tbf],size=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#-%%ProdG6$%\"ZG-%)SequenceG6#-F%6%%(EpsilonGF'F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "expleT:=draw([T, tbf], size= 20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'expleTG-%%ProdG6$%\"ZG-%)Se quenceG6#-F&6%-F*6#-F&6%%(EpsilonGF(-F*6#-F&6%-F*6#-F&6%F2F(-F*6#-F&6% -F*6$-F&6%-F*6#-F&6%F2F(-F*6#-F&6%F2F(-F*6$-F&6%F2F(-F*6#-F&6%-F*6#-F& 6%F2F(-F*6#-F&6%F2F(F2F(F2-F&6%-F*6$FYFYF(F2F(F2-F&6%FSF(FWF(F2F(F2F(F 2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "We can also use gfseries to \+ retrieve the first terms of the generating functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Order:=10:gfseries(tbf,unlabelled,z );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7&/-%\"FG6#%\"zG+9F+ \"\"\"\"\"!F-\"\"\"\"\"$\"\"#\"#7\"\"$\"#b\"\"%\"$t#\"\"&\"%G9\"\"'\"% _x\"\"(\"&jK%\"\")\"'vmC\"\"*-%\"OG6#F-\"#5/-%\"BGF*+7F+F-\"\"\"\"\"# \"\"#\"\"(\"\"$\"#I\"\"%\"$V\"\"\"&\"$G(\"\"'\"%wQ\"\"(\"&=8#\"\")\"'v ,7\"\"*F@\"#5/-%\"TGF*+7F+F-\"\"\"F-\"\"#F0\"\"$F2\"\"%F4\"\"&F6\"\"'F 8\"\"(F:\"\")F<\"\"*F@\"#5/-%\"ZGF*+%F+F-\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "And here is an example of random tree:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plotTree(draw([T,tbf],size=20));" } }{PARA 13 "" 1 "" {INLPLOT "67-%'CURVESG6#7$7$$!3`P:&H;l0^*!#=$!3^u%\\ P%*p,4$F*7$$!3^u%\\P%*p,4)F*$!3PJZ#H__y(eF*-F$6#7$7$F($\"3^u%\\P%*p,4$ F*7$F.$\"3PJZ#H__y(eF*-F$6#7$7$F6$\"3`P:&H;l0^*F*7$F+F?-F$6#7$F-7$F0F. -F$6#7$FEF5-F$6#7$7$F?F+F5-F$6#7$7$$\"3^u%\\P%*p,4)F*F0FL-F$6#7$7$F+F( FL-F$6#7$7$F6F(FV-F$6#7$7$F9F.FP-F$6#7$7$\"\"!$!\"\"F]oFV-F$6#7$7$F]o$ \"\"\"F]oF>-F$6#7$FA7$F0FQ-F$6#7$FioF8-F$6#7$7$FQF97$FdoF]o-F$6#7$7$F? F6Fap-F$6#7$7$F^oF]oF'-F$6#7$7$F9FQF`p-F$6#7$F8Fap-%*AXESSTYLEG6#%%NON EG-%'COLOURG6&%$RGBGF]oF]o$\"*++++\"!\")" 2 263 274 274 2 0 1 0 2 6 0 1 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 255 0 0 255 1 0 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "Forests" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "In order to dissociate the trees in a fo rest, let us substitute to each vertex of a tree another vertex togeth er with a forest" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "fo:=\{ B=Prod(Sequence(B),V,Sequence(B)),\n V=Union(Prod(Z, F)),\n F= Union(Epsilon, Prod(V, Sequence(B)))\}; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#foG<%/%\"BG-%%ProdG6%-%)SequenceG6#F'%\"VGF+/F.-%&UnionG6#-F) 6$%\"ZG%\"FG/F6-F16$%(EpsilonG-F)6$F.F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "seq(count([F, fo], size=i), i=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"\"\"\"#\"\"(\"#L\"$\"=\"%$3\"\"%ao\"&6^%\"'HcI \"($G<@" }}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 28 "Connected and gener al graphs" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 16 "Connected graphs" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "To see how nc-graphs are built, co nsider the set of childs " }{XPPEDIT 18 0 "v[i],v[i+1],`...`,v[j]" "6& &%\"vG6#%\"iG&F$6#,&F&\"\"\"\"\"\"F*%$...G&F$6#%\"jG" }{TEXT -1 22 " a ttached to the root " }{XPPEDIT 18 0 "v[1]" "&%\"vG6#\"\"\"" }{TEXT -1 22 ". The graphs built on " }{XPPEDIT 18 0 "v[2],`...`,v[i]" "6%&% \"vG6#\"\"#%$...G&F$6#%\"iG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v[j], `...`,v[n-1]" "6%&%\"vG6#%\"jG%$...G&F$6#,&%\"nG\"\"\"\"\"\"!\"\"" } {TEXT -1 0 "" }{TEXT -1 67 " are connected by hypothesis, while betwee n any two other vertices " }{XPPEDIT 18 0 "v[k]" "&%\"vG6#%\"kG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "v[k+1]" "&%\"vG6#,&%\"kG\"\"\"\"\" \"F'" }{TEXT -1 67 " one can have either one connected graph or two co nnected graphs. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 254 "But any graph built from a sequence of successive vertic es of the circle is a system of arches since the arcs which are chords are not allowed to cross. The endpoints of these arches are shared by the graphs built to the left and to the right of a given " }{XPPEDIT 18 0 "v[k]" "&%\"vG6#%\"kG" }{TEXT -1 57 ", so that we shall say that \+ the size of an arch built on " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 0 "" }{TEXT -1 11 " points is " }{XPPEDIT 18 0 "n-1" ",&%\"nG\"\"\" \"\"\"!\"\"" }{TEXT -1 186 ". At last, we are interested here in Eleme ntary Arches, that is arches that always contain an arc between the fi rt and last points. General arches are easily obtained by sequencing E As. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 165 " From this discussion we derive the Combstruct specification of NC grap hs containing at least 2 vertices. In particular, the 5 arguments of a C entity are as follows:" }}{PARA 0 "" 0 "" {TEXT -1 32 "1.first Z: t he root of the graph" }}{PARA 0 "" 0 "" {TEXT -1 45 "2.second Z: the l efmost point of the first EA" }}{PARA 0 "" 0 "" {TEXT -1 30 "3 and 4. \+ the two EAs built on " }{XPPEDIT 18 0 "v[2],`...`,v[i]" "6%&%\"vG6#\" \"#%$...G&F$6#%\"iG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v[j],`...`,v[ n-1]" "6%&%\"vG6#%\"jG%$...G&F$6#,&%\"nG\"\"\"\"\"\"!\"\"" }}{PARA 0 " " 0 "" {TEXT -1 62 "5.the sequence of EAs found between two consecutiv e childs of " }{XPPEDIT 18 0 "v[1]" "&%\"vG6#\"\"\"" }{TEXT -1 0 "" } {TEXT -1 39 ". The first term corresponds to one EA," }}{PARA 0 "" 0 " " {TEXT -1 116 "and the second one to two EAs. For the latter case, th e Z in the Prod stands for the lefmost point of the secong EA." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 299 "ar:=\{EA = Union(Sequence(E A, card >= 2), \n Prod(Z, Sequence(EA), Sequence(EA))\n \+ ),\n C=Union(Z,\n Prod(Z,Z,Sequence(EA), \+ Sequence(EA),\n Sequence(Union(Sequence(EA,card>=1), \+ Prod(Z,Sequence(EA),Sequence(EA))))))\}; \n\n \+ " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#arG<$/%#EAG-%&UnionG6$-%)Seque nceG6$F'1\"\"#%%cardG-%%ProdG6%%\"ZG-F,6#F'F5/%\"CG-F)6$F4-F26'F4F4F5F 5-F,6#-F)6$-F,6$F'1\"\"\"F0F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "We can now count the number of elementary arches of a given size. The sequence found is not in [Sloa95]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "seq(count([EA,ar], size=i),i=1..20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "66\"\"\"\"\"$\"#;\"$0\"\"$o(\"%1g\"&_\"\\\"',dT\"(![ /O\")5/(=$\"*[7E'G\"+!p\"o/E\",[W'y&R#\"-+Vu*RA#\".7\\l643#\"/X+YPXg> \"0[o7qJ\"e=\"1q(GX\")e0x\"\"2+G&Q#pP^p\"\"3qYnU#)f%)H;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "allstructs([EA,ar],size=2);" }} {PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%-%) SequenceG6$-%%ProdG6%%\"ZG%(EpsilonGF+F'-F(6%F*-F%6#F'F+-F(6%F*F+F." } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "We can also count the number of NC-graphs. The corresponding sequence turns out to be M3594 in [Sloa9 5] and gives the reverse " }}{PARA 0 "" 0 "" {TEXT -1 24 "of the g.f. \+ for squares:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "seq(count([ C,ar], size=i),i=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"\"F#\" \"%\"#B\"$c\"\"%i6\"%#>*\"&>e(\"'3\\k\"(#=;c" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 83 "As usual, we can get all the structures of a given size , or just draw some of them:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "allstructs([C,ar],size=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7& -%%ProdG6'%\"ZGF'%(EpsilonGF(-%)SequenceG6#-F*6#-F%6%F'F(F(-F%6'F'F'F( F(F,-F%6'F'F'F(F,F(-F%6'F'F'F,F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "draw([EA,ar],size=10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%)SequenceG6$-F$6%-%%ProdG6%%\"ZG%(EpsilonGF,-F)6%F+F,-F$6#-F)6 %F+F,-F$6#-F)6%F+F,-F$6#F(F(-F$6$-F$6%F(F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "And an example of random graph:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 34 "plotNCGraph(draw([C,ar],size=10));" }}{PARA 13 "" 1 "" {INLPLOT "61-%'CURVESG6#7$7$$!3^u%\\P%*p,4$!#=$\"3`P:&H;l0^ *F*7$$!3^u%\\P%*p,4)F*$\"3PJZ#H__y(eF*-F$6#7$F-7$$\"3^u%\\P%*p,4)F*$!3 PJZ#H__y(eF*-F$6#7$7$$\"3^u%\\P%*p,4$F*$!3`P:&H;l0^*F*F5-F$6#7$F-7$F.F 8-F$6#7$7$$!\"\"\"\"!FLFE-F$6#7$F-7$F(F@-F$6#7$7$F6F0F5-F$6#7$F-FI-F$6 #7$FPF=-F$6#7$FT7$F>F+-F$6#7$FhnF'-F$6#7$F'F5-F$6#7$7$$\"\"\"FLFLF5-%* AXESSTYLEG6#%%NONEG-%'COLOURG6&%$RGBGFLFL$\"*++++\"!\")" 2 274 286 286 2 0 1 0 2 6 0 1 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 255 0 0 255 1 0 0 0 0 0 -6016 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 14 "Gen eral graphs" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 343 "As observed in [Fl aNo97], a general graph is obtained from a connected one by the substi tution Z -> Prod(Z, G). So that we just have to rewrite the previous \+ grammar by adding a new symbol which makes this substitution. Notice h owever that the decomposition of a connected graph misses a configurat ion for general graphs: the one where vertex " }{XPPEDIT 18 0 "v[1]" " &%\"vG6#\"\"\"" }{TEXT -1 49 " does not have any child, which we there fore add:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 384 "br:=\{EA = Un ion(Sequence(EA, card >= 2), \n Prod(V, Sequence(EA), S equence(EA))\n ),\n V=Union(Prod(Z, G)),\n G=Uni on(Epsilon,\n Prod(Z, G), \n Prod(V,V,Sequence (EA), Sequence(EA),\n Sequence(Union(Sequence(EA,card >=1), Prod(V,Sequence(EA),Sequence (EA)))))\n )\n \};" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% #brG<%/%#EAG-%&UnionG6$-%)SequenceG6$F'1\"\"#%%cardG-%%ProdG6%%\"VG-F, 6#F'F5/F4-F)6#-F26$%\"ZG%\"GG/F=-F)6%%(EpsilonGF:-F26'F4F4F5F5-F,6#-F) 6$-F,6$F'1\"\"\"F0F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "The numbe r of graphs is given by the following sequence, not to be found in [Sl oa95]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ggSeq:=[seq(count ([G, br], size=i), i=0..20)];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&gg SeqG77\"\"\"F&\"\"#\"\")\"#[\"$_$\"%!)G\"&;_#\"'o6B\"([3>#\")K?H@\"*_V /6#\"+kkCD@\",;[&>o@\"-'ppIiB#\".kW<=yK#\"/%=hC%[UC\"0o$)p_Z0e#\"1K)oP GkHu#\"2#fiChCCJH\"3)oarGr*RZJ" }}}{EXCHG {PARA 12 "" 1 "" {TEXT -1 75 "In this case, it also turns out that the differential equation ver ified by " }{XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 15 " is of order 1: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "ggDiffEq:=listtodiffeq( ggSeq,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)ggDiffEqG7$<$/-%\"y G6#\"\"!\"\"\",*!\"\"F,%\"xG\"#=*&,(F,F,F/!#=*$F/\"\"#\"\")F,-F)6#F/F, F,*&,(F4\"#7*$F/\"\"$!\"%F/F.F,-%%diffG6$F7F/F,F,%$ogfG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "From this equation and the condition " } {XPPEDIT 18 0 "coeff(y(x), x, 1)=1" "/-%&coeffG6%-%\"yG6#%\"xGF)\"\"\" \"\"\"" }{TEXT -1 35 ", we get the following closed form:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "closedForm:=dsolve(ggDiffEq[1],y(x) );\nsubs(_C1=-1/2,closedForm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+c losedFormG/-%\"yG6#%\"xG,*\"\"\"F+F)#\"\"$\"\"#*$F)F.!\"\"*(F)F+,(F)!# 7F/\"\"%F+F+#F+F.%$_C1GF+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG 6#%\"xG,*\"\"\"F)F'#\"\"$\"\"#*$F'F,!\"\"*&F'F),(F'!#7F-\"\"%F)F)#F)F, #F.F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "As shown in [FlaNo97], t his corresponds to the general term:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "cn:=proc(n)\nlocal k,l; \nsum((-1)^k*product(2*l-1,l =1..n-k-1)/(factorial(k)*factorial(n-2*k))*3^(n-2*k)*2^(-k-2), k=0..iq uo(n,2))\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "gn:=proc( n) 2^n*cn(n-1) end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#gnG:6#%\"nG6 \"F(F(*&)\"\"#9$\"\"\"-%#cnG6#,&F,F-!\"\"F-F-F(F(" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 20 "seq(gn(i), i=3..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*\"\")\"#[\"$_$\"%!)G\"&;_#\"'o6B\"([3>#\")K?H@" }}}}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 26 "Dissections and partitions" }} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 11 "Dissections" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "A dissection of a convex polygon " }{XPPEDIT 18 0 "P [n]=\{v[1],`...`,v[n]" "/&%\"PG6#%\"nG<%&%\"vG6#\"\"\"%$...G&F)6#F&" } {TEXT -1 0 "" }{TEXT -1 135 " is a partition of the polygon into polyg onal regions by means of non-crossing diagonals. If the polygonal reg ion containing the edge " }{XPPEDIT 18 0 "v[1]*v[2]" "*&&%\"vG6#\"\"\" \"\"\"&F$6#\"\"#F'" }{TEXT -1 0 "" }{TEXT -1 5 " has " }{XPPEDIT 18 0 "r+1" ",&%\"rG\"\"\"\"\"\"F$" }{TEXT -1 54 " sides, one gets a bigger \+ dissection by replacing the " }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 0 "" }{TEXT -1 492 " edges by a dissection. So that a dissection is ei ther an edge connecting two vertices or a sequence of dissections. The tricky point in sequencing 2 dissections consists in not counting the same vertex twice, as for the connected graphs above. When sequencing dissections, we therefore assume that each dissection provides its ri ghtmost point while the lefmost one is the rightmost point of the diss ection to the left in the sequence. With this convention, the number o f dissections of size " }{XPPEDIT 18 0 "i" "I\"iG6\"" }{TEXT -1 0 "" } {TEXT -1 61 " actually counts the number of dissections of a polygon w ith " }{XPPEDIT 18 0 "i+1" ",&%\"iG\"\"\"\"\"\"F$" }{TEXT -1 0 "" } {TEXT -1 30 " vertices, and the grammar is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "dissG:=\{Di=Union(Z, Sequence(Di, card >= 2))\};\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&dissGG<#/%#DiG-%&Un ionG6$%\"ZG-%)SequenceG6$F'1\"\"#%%cardG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "The corresponding sequence, M2898 in [Sloa95] and related to Schroeder's second problem is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "seq(count([Di, dissG], size=i), i=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"\"F#\"\"$\"#6\"#X\"$(>\"$.*\"%zU\"&$z?\" '\\I5" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 11 "Partitions " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "A non-crossing partition of size " } {XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 0 "" }{TEXT -1 48 " is a partit ion of [n]=\{1,2,...,n\} such that if " }{TEXT -1 29 "a " 0 "" {MPLTEXT 1 0 38 "partG:=\{P=Sequence(V), V = Prod(Z,P)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&partGG<$/%\"PG-%)SequenceG6#%\"VG/F+-%%ProdG 6$%\"ZGF'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "We get the sequence \+ of Catalan numbers, which can be checked in terms of generating functi ons:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "seq(count([P, partG ], size=i), i=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"\"\"\"#\" \"&\"#9\"#U\"$K\"\"$H%\"%I9\"%i[\"&'z;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "gfsolve(partG, unlabelled, z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%\"ZG6#%\"zGF(/-%\"VGF',&#\"\"\"\"\"#F.*$,&F.F.F(! \"%F-#!\"\"F//-%\"PGF',$*&F(F4,&F.F.F0F4F.F-" }}}}}}{SECT 0 {PARA 3 " " 0 "" {TEXT -1 22 "Univariate asymptotics" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 30 "Asymptotic counting: algorithm" }}{SECT 0 {PARA 5 "" 0 " " {TEXT -1 7 "Outline" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "As shown \+ in [FlaNo97] for the six NC configurations we are interested in, the g enerating function " }{XPPEDIT 18 0 "y(z)" "-%\"yG6#%\"zG" }{TEXT -1 81 " satisfies an algebraic equation. In the case of NC forests for ex ample, we have:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eq:=y^3+ (-z+z^2-3)*y^2+(z+3)*y-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG,** $%\"yG\"\"$\"\"\"*&,(%\"zG!\"\"*$F,\"\"#F)!\"$F)F)F'F/F)*&,&F,F)F(F)F) F'F)F)F-F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Like in many implic itely defined functions [Dr97, HaPa73], we expect " }{TEXT 258 9 "a pr iori " }{XPPEDIT 18 0 "y(z) " "-%\"yG6#%\"zG" }{TEXT -1 63 " to have l ocally an expansion of the square-root type, that is:" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "y(z)=c[0]+c[1]*sqrt(1-z/rho) +c[2]*(1-z/rho)+O((1- z/rho)^(3/2))" "/-%\"yG6#%\"zG,*&%\"cG6#\"\"!\"\"\"*&&F)6#\"\"\"F,-%%s qrtG6#,&\"\"\"F,*&F&F,%$rhoG!\"\"F8F,F,*&&F)6#\"\"#F,,&\"\"\"F,*&F&F,F 7F8F8F,F,-%\"OG6#),&\"\"\"F,*&F&F,F7F8F8*&\"\"$F,\"\"#F8F," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "By a singularity \+ analysis at the dominant singularity " }{XPPEDIT 18 0 "rho" "I$rhoG6\" " }{TEXT -1 14 " and denoting " }{XPPEDIT 18 0 "gamma=-c[1]/2" "/%&gam maG,$*&&%\"cG6#\"\"\"\"\"\"\"\"#!\"\"F," }{TEXT -1 9 ", we get:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "[z^n] *y(z) = c[1]/\\Gamma(-1/2)* rho^(-n)/sqrt(n^3) * (1+O(1/n)) " "/*&7#)% \"zG%\"nG\"\"\"-%\"yG6#F&F(*,&%\"cG6#\"\"\"F(-%&GammaG6#,$*&\"\"\"F(\" \"#!\"\"F8F8)%$rhoG,$F'F8F(-%%sqrtG6#*$F'\"\"$F8,&\"\"\"F(-%\"OG6#*&\" \"\"F(F'F8F(F(" }{TEXT -1 5 " or " }{XPPEDIT 18 0 "y[n]=gamma*rho^(-n )/sqrt(Pi*n^3)*(1+O(1/n))" "/&%\"yG6#%\"nG**%&gammaG\"\"\")%$rhoG,$F&! \"\"F)-%%sqrtG6#*&%#PiGF)*$F&\"\"$F)F-,&\"\"\"F)-%\"OG6#*&\"\"\"F)F&F- F)F)" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "We now e xamplify this method for the class of NC-forests. The singulatities so ught may arise at those points " }{XPPEDIT 18 0 "z" "I\"zG6\"" }{TEXT -1 24 " such that E(z,y)=0 and " }{XPPEDIT 18 0 "diff(E(z,y),y)=0" "/- %%diffG6$-%\"EG6$%\"zG%\"yGF*\"\"!" }{TEXT -1 33 ". In order to get th e candidates " }{XPPEDIT 18 0 "z" "I\"zG6\"" }{TEXT -1 57 " satisfynin g these conditions, we just have to eliminate " }{XPPEDIT 18 0 "y" "I \"yG6\"" }{TEXT -1 68 " between the two previous equations through a r esultant computation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "r es_y:=resultant(eq,diff(eq,y),y);\nres_y:=expand(normal(res_y/gcd(res_ y,diff(res_y,z))));\nOmega[0]:=normal(res_y/z^ldegree(res_y));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&res_yG,**$%\"zG\"\"$!\"%*$F'\"\"%\" #K*$F'\"\"&\"\")*$F'\"\"'!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&re s_yG,*%\"zG!\"%*$F&\"\"#\"#K*$F&\"\"$\"\")*$F&\"\"%!\"&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%&OmegaG6#\"\"!,*!\"%\"\"\"%\"zG\"#K*$F+\"\"# \"\")*$F+\"\"$!\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 354 "And the si ngularities sought are the solutions of the previous equation whose mo dulus is smaller than 1. If the set found has cardinality one, we are \+ done. It actually turns out that this is the case for all our configur ations but the connected graphs where a `ghost' singularity has to be \+ elimanated by an external argument --see [FlaNo97]. In our case:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "rhonums:=[fsolve(Omega[0],z, complex)];\nOmega[1]:=map(proc(x) if abs(x)<1 then x fi end,rhonums); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(rhonumsG7%$!+2LGI>!\"*$\"+p5&e@ \"!#5$\"++#)p3MF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&OmegaG6#\"\" \"7#$\"+p5&e@\"!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "And a conve nient way to represent our singularity both in symbolic and numerical \+ form is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "rho_symb:=RootO f(Omega[0], z, op(Omega[1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)rh o_symbG-%'RootOfG6$,*\"\"%\"\"\"%#_ZG!#K*$F+\"\"#!\")*$F+\"\"$\"\"&$\" +p5&e@\"!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Since in our case \+ the dominant coefficient of the equation in " }{XPPEDIT 18 0 "y" "I\"y G6\"" }{TEXT -1 76 " does not vanish, the function remains finite at i ts singularity. Its value " }{XPPEDIT 18 0 "tau=Limit(y(z),z=rho)" "/% $tauG-%&LimitG6$-%\"yG6#%\"zG/F*%$rhoG" }{TEXT -1 22 " is also the qua ntity " }{XPPEDIT 18 0 "c[0]" "&%\"cG6#\"\"!" }{TEXT -1 32 " defined a bove. Since the point " }{XPPEDIT 18 0 "P(rho,tau)" "-%\"PG6$%$rhoG%$t auG" }{TEXT -1 29 " is a singularity, we have " }{XPPEDIT 18 0 "E(rh o,tau)=0" "/-%\"EG6$%$rhoG%$tauG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "diff(E(z,y),y)[rho,tau]=0" "/&-%%diffG6$-%\"EG6$%\"zG%\"yGF+6$%$ rhoG%$tauG\"\"!" }{TEXT -1 23 ". Candidate values for " }{XPPEDIT 18 0 "tau" "I$tauG6\"" }{TEXT -1 35 " are therefore obtained as follows: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "deq:=subs(z=rho_symb, \+ diff(eq,y));\nprint(`Candidate values for function at singularity`);\n tau_vals:=[fsolve(deq,y)];\nprint(tau_vals);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$deqG,**$%\"yG\"\"#\"\"$*&,(-%'RootOfG6$,*\"\"%\"\"\" %#_ZG!#K*$F2F(!\")*$F2F)\"\"&$\"+p5&e@\"!#5!\"\"*$F,F(F1!\"$F1F1F'F1F( F,F1F)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%MCandidate~values~for~fun ction~at~singularityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)tau_valsG7 $$\"+?q\")o&)!#5$\"+W(>V@\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$ \"+?q\")o&)!#5$\"+W(>V@\"!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "Plugging back these candidates into the equation eq, we could get the correct one from a carefully controlled numerical analysis:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "eTau:=subs(z=rho_symb, eq); \n[seq( evalf(subs(y=tau_vals[i], eTau)), i=1..nops(tau_vals))];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eTauG,**$%\"yG\"\"$\"\"\"*&,(-%'Roo tOfG6$,*\"\"%F)%#_ZG!#K*$F1\"\"#!\")*$F1F(\"\"&$\"+p5&e@\"!#5!\"\"*$F, F4F)!\"$F)F)F'F4F)*&,&F,F)F(F)F)F'F)F)F;F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+P!\\LG#!#6$!#pF&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "To get the constant " }{XPPEDIT 18 0 "c[1]" "&%\"cG6#\"\" \"" }{TEXT -1 72 " we are missing, let us just compute the Puiseux exp ansions verified by " }{XPPEDIT 18 0 "y(z)" "-%\"yG6#%\"zG" }{TEXT -1 56 " with the algeqtoseries procedure from the gfun package:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "all_puis:=algeqtoseries(subs (z=rho_symb*(1-t^2),eq),t,y,6);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%) all_puisG7$++%\"tG,(#\"#D\"#P\"\"\"-%'RootOfG6$,*\"\"%F,%#_ZG!#K*$F2\" \"#!\")*$F2\"\"$\"\"&$\"+p5&e@\"!#5#F,F+*$F-F5#!\"#F+\"\"!,$**,(!#DF,F -!\"\"F>F5F,F-F,,*!#7F,F-!#\\F>!\"%*$F-F8F1F,,,F-\"$`&!$K%F,F>!%B>FK!$ 5#*$F-F1\"$O\"FFFF\"\"#,$**FDF,F-F5,8F>!)cZ=iF-!)QMsL*$F-\"\"*!'oH:*$F -\"\")!(+p[\"*$F-\"\"(\"(1Vq$*$F-\"\"'\"))y%e=*$F-F9!)'o55#FK!)=\"F,F,FL!\"$F@\"\"%-%\"OG6#F,\"\"',0#\"#VF+F ,F-#\"#=F+F>#!#N\"#u*&-F.6#,*F4\"%p8F>\"%!H&!$G#F,F-\"$\")*F,F'F,F,*&, (F>#\"%l8Fip#!#)*FipF,F-#!$k'FipF,F'F5F,*&,(*&FfpF,F-F,#\"%dY\"&`1&*&F -F5FfpF,#!%q^FjqFfp#!&T[#FjqF,F'F8F,-Fjo6#*$F'#FjnF5F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "and select those with a " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 82 " term --which corresponds to the square-r oot sought due to the change of variable " }{XPPEDIT 18 0 "z=rho*(1-t^ 2)" "/%\"zG*&%$rhoG\"\"\",&\"\"\"F&*$%\"tG\"\"#!\"\"F&" }{TEXT -1 1 ": " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 252 "#--convert into series i.e. remove the O()\nall_puis_s:=map(eval, map2(subs, O=0, all_puis)) ;\n#--collect the coeff in t and select the non-null one(s)\nc1:=map(p roc(x) if x<>0 then x fi end, map(coeff, all_puis_s, t, 1));\nprint(`C onstant`, evalf(c1,10));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+all_pui s_sG7$+)%\"tG,(#\"#D\"#P\"\"\"-%'RootOfG6$,*\"\"%F,%#_ZG!#K*$F2\"\"#! \")*$F2\"\"$\"\"&$\"+p5&e@\"!#5#F,F+*$F-F5#!\"#F+\"\"!,$**,(!#DF,F-!\" \"F>F5F,F-F,,*!#7F,F-!#\\F>!\"%*$F-F8F1F,,,F-\"$`&!$K%F,F>!%B>FK!$5#*$ F-F1\"$O\"FFFF\"\"#,$**FDF,F-F5,8F>!)cZ=iF-!)QMsL*$F-\"\"*!'oH:*$F-\" \")!(+p[\"*$F-\"\"(\"(1Vq$*$F-\"\"'\"))y%e=*$F-F9!)'o55#FK!)=\"F,F,FL!\"$F@\"\"%,.#\"#VF+F,F-#\"#=F+F>#!#N\" #u*&-F.6#,*F4\"%p8F>\"%!H&!$G#F,F-\"$\")*F,F'F,F,*&,(F>#\"%l8Fep#!#)*F epF,F-#!$k'FepF,F'F5F,*&,(*&FbpF,F-F,#\"%dY\"&`1&*&F-F5FbpF,#!%q^FfqFb p#!&T[#FfqF,F'F8F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#c1G7#-%'RootO fG6#,**$%#_ZG\"\"#\"%p8*$-F'6$,*\"\"%\"\"\"F+!#KF*!\")*$F+\"\"$\"\"&$ \"+p5&e@\"!#5F,\"%!H&!$G#F3F/\"$\")*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%)ConstantG7#$!+mb=$\\\"!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 " Should we have found several expansions containing a " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 7 " term, " }{TEXT -1 58 "the right one cou ld have been selected from the formula " }{XPPEDIT 18 0 "c[1]=-sqrt( 2*rho*p[1,0]/p[0,2])" "/&%\"cG6#\"\"\",$-%%sqrtG6#**\"\"#\"\"\"%$rhoGF -&%\"pG6$\"\"\"\"\"!F-&F06$F3\"\"#!\"\"F7" }{TEXT -1 6 " and " } {XPPEDIT 18 0 "p[i,j]=Diff(eq,z$i,y$j)[rho,tau]/(factorial(i)*factoria l(j)" "/&%\"pG6$%\"iG%\"jG*&&-%%DiffG6%%#eqG-%\"$G6$%\"zGF&-F/6$%\"yGF '6$%$rhoG%$tauG\"\"\"*&-%*factorialG6#F&F8-F;6#F'F8!\"\"" }{TEXT -1 54 ". Also note that the minus sign has to be adopted for " }{XPPEDIT 18 0 "c[1]" "&%\"cG6#\"\"\"" }{TEXT -1 0 "" }{TEXT -1 59 " since the g enerating function increases with its argument." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "We can locally plot the algebraic curve in the neigh borhood of the singularity:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "implicitplot(eq, z=-.1..0.15, y=-1..1.3, numpoints=5000);" }} {PARA 13 "" 1 "" {INLPLOT "6$-%'CURVESG6e^l7$7$$!31+++++++5!#=$\"3X2lP 1Io_\"*F*7$$!3eXK9P#pFr*!#>$\"3*[k\"[;uOs\"*F*7$7$$!3hhO%RK()yk*F0$\"3 6\\]qs?>z\"*F*F-7$F47$$!3-,$p()[5oQ*F0$\"3nY%\\*H!Hk>*F*7$7$$!3WDt)yku dH*F0$\"3anmV*)F0$\"3[wV`2PmK#*F*FI7$FO7$$!3XDXe2BhN()F 0$\"3p9J?FS@X#*F*7$7$$!3aZYx&H\\:f)F0$\"3)fi$R0>ff#*F*FU7$Fen7$$!3l\\n ?4*p.T)F0$\"3S$RnEoM*p#*F*7$7$$!3)*4$=(>mVR#)F0$\"3bAZK*)*>mG*F*F[o7$F ao7$$!3=r'R$H=O&3)F0$\"3K6:;64([H*F*7$7$$!3Vs>mVRK()yF0$\"3B2\")\\o;s8 $*F*Fgo7$F]p7$$!3^L1'*HSegxF0$\"3$)GYq(**=+K*F*7$7$$!3[Lcgn7@NvF0$\"3. ;p=1d'3M*F*Fcp7$Fip7$$!3EzN#)35.OuF0$\"3')p.B!)QPX$*F*7$7$$!3#fH\\:f)4 $=(F0$\"3i:Y:I\\,o$*F*F_q7$Feq7$$!3M%zt([ap6rF0$\"3R)Rcp#)G4P*F*7$7$$! 3OeH\\:f)4$oF0$\"3^'*oh.]7&R*F*F[r7$Far7$$!38N#*enxc(y'F0$\"3@L%QA-vmR *F*7$7$$!3T>mVRK()ykF0$\"3h2%on)G9A%*F*Fgr7$F]s7$$!3))*HeS`NOY'F0$\"3b /=OH5gA%*F*7$7$$!3fG9RBux%G'F0$\"3\"*f)4$=(>mV*F*Fcs7$Fis7$$!39G[1Z1eR hF0$\"3WT'*H&>9%[%*F*7$7$$!3&=G!Qj0wEhF0$\"3K.%p&)R?-X*F*F_t7$Fet7$$!3 \"\\\\S7(QY:eF0$\"3D&38NUqTZ*F*7$7$$!3IWRK()ykudF0$\"3IrfRJp&)z%*F*F[u 7$7$$!3gVRK()ykudF0$\"3>qfRJp&)z%*F*7$$!3G/$yXK!p\"\\&F0$\"3]-dbACC+&* F*7$7$$!3N0wE6_`AaF0$\"3C\"3EkxY(4&*F*F\\v7$7$$!3l/wE6_`AaF0Fev7$$!3B] @8!fs#o^F0$\"3#>2P)oq&* F*Fgw7$7$$!3BI\\:f)4$=ZF0F`x7$$!3f4%Q;)ocAXF0$\"3_\"\\m4%)z/e*F*7$7$$! 3G\"f)4$=(>mVF0$\"33t;#y%on+'*F*Ffx7$7$F]y$\"3>u;#y%on+'*F*7$$!38*=$e( *)=.?%F0$\"3^?vgd`&zg*F*7$7$$!3s`A/2X39SF0$\"32fo)R$)o7j*F*Fey7$F[z7$$ !3'H6]nx4&yQF0$\"3Q?HTGY$ej*F*7$7$$!3Z:f)4$=(>m$F0$\"3CPA5)Q\"*=m*F*Fa z7$Fgz7$$!3m&*y\"*Qo:k'*F*7$7$$!3Ax&H\\:f)4LF0$\"3quw $G[PCp*F*F][l7$Fc[l7$$!3sm/`#QrjB$F0$\"3Q3XnHY&Hp*F*7$7$$!3mRK()ykudHF 0$\"3>^!yNp`Fs*F*Fi[l7$F_\\l7$$!3u^.]&>nh\"HF0$\"3F0$\"3C%=e4D:`!)*F*7$7$$!3DDUq]%39!>F0$\"3Ihw# )H*Rx\")*F*F\\_l7$Fb_l7$$!3Tg!\\h0odi\"F0$\"3RpFEzy\"4$)*F*7$7$$!3M()y kudH\\:F0$\"3*y!)GiTU:&)*F*Fh_l7$F^`l7$$!3Q)yy'fEu+8F0$\"3E]>[w\"Qe&)* F*7$7$$!3E\\:f)4$=(>\"F0$\"3f\\)=e$p+'))*F*Fd`l7$Fj`l7$$!3MkVy*p!GQ(*! #?$\"37e%HfT?!z)*F*7$7$$!3$=6_`A/2X)Fcal$\"3FW$z`7y8#**F*F`al7$Fgal7$$ !3p?9;L:N8kFcal$\"3Cx>d%fqq*)*F*7$7$$!3xK()ykudH\\Fcal$\"3b:r\"p&f^e** F*F]bl7$7$Fdbl$\"3m;r\"p&f^e**F*7$$!3+c>4yzT9GFcal$\"39_ `A/2X39FcalFecl7$7$Fjcl$\"3'=Fcal$\"3>xHx#o))=/\"Fgcl7$7$Fjcl$\"3b\"R(> px.V5FgclFefl7$7$$\"3v39!p\"G!Qj&Fcal$\"3wrD/f0Wj`J Fcal$\"3!)e)4$=(>mV*F*7$7$$\"3+\"=&>0Wj`JFcal$\"3pd)4$=(>mV*F*Ficl7$7$ Fjcl$\"33sTm*y`0+\"Fgcl7$$\"3>%4I.)R!RR#Fcal$\"34DYK2M'e+\"Fgcl7$7$F`g l$\"3&H))\\CT\\a+\"FgclFchl7$F[gl7$$\"3(G#)=00njK%Fcal$\"3$[Dt%zM(G0\" Fgcl7$7$$\"34())f`997W&Fcal$\"3@=(>mVRK2\"FgclF]il7$Fcil7$$\"3[4>/*e2T e&Fcal$\"3cDgrKmpt5Fgcl7$7$F`gl$\"3@RFA$=)*R2\"FgclFiil7$7$$\"3o)ykudH \\:*Fcal$\"3ST*Q0\"yBq!*F*7$$\"3qBdvBSsgzFcal$\"3CG!Qj0wE6*F*7$FhjlF_g l7$Fihl7$$\"3MQDwr'pU\"eFcal$\"3nH1;4/z15Fgcl7$7$$\"3i!ySZA+_&zFcalF`e lF_[m7$Fe[m7$$\"3$z&3Tl')yk*)Fcal$\"3:[ih++?55Fgcl7$7$Fdjl$\"3&e*z-GY^ 45FgclFi[m7$F_jl7$$\"3B5)f$))*3&G\")Fcal$\"3-%[Rg]#o#3\"Fgcl7$7$$\"3T! zkudH\\:*Fcal$\"3-$H]<#e5!4\"FgclFc\\m7$Fcjl7$$\"3Q4*ouD2*45F0$\"3)HR$ f+\\\"e-*F*7$7$$\"3&p\"G!Qj0wE\"F0$\"30Yh*>dkv\"*)F*F_]m7$F_\\m7$$\"3c $=c9%zt)>\"F0$\"3NU7#>`'y95Fgcl7$7$Ff]m$\"3p?JL:Lf85FgclF[^m7$7$FdjlF \\]m7$$\"3)H,jWqxQ.\"F0$\"3L!)*=hSUZ4\"Fgcl7$7$$\"3)>%H2)*3r]6F0$\"3\\ ,p\"G!Qj06FgclFf^m7$F\\_m7$$\"3.9cY9#=7C\"F0$\"36CfRX9136Fgcl7$7$Ff]m$ \"3n/=da!=#46FgclFb_m7$7$$\"3&[:f)4$=(>;F0$\"3pi*eoVUev)F*7$$\"3QOw;?/ R.:F0$\"3e'>mVRK()y)F*7$7$$\"3cOw;?/R.:F0$\"3p(>mVRK()y)F*Fe]m7$Fa^m7$ $\"3sG=5Fgcl7$7$F]`m$\"3B5[Q&f^u,\"FgclF]am7$7$ $\"37D@5FgclFccm7$F`bm7$$ \"3#fc)f%o-\"fx9T2LO6FgclF]dm7$F]c m7$$\"3!4R4p/!pMAF0$\"3#=\"=De?!pa)F*7$7$$\"3,J=(>mVRK#F0$\"3\")eq^B#Q k^)F*Fgdm7$Ficm7$$\"3fjBD`u;Z@F0$\"3)HaFa69Z-\"Fgcl7$7$F^em$\"3&\\(opl )\\]-\"FgclFcem7$7$F^cm$\"3[>x9T2LO6Fgcl7$$\"3f`$)oN\")p&)>F0$\"3H\") \\B\"R_n8\"Fgcl7$7$$\"3Znf.]P?2?F0$\"3b%39!p\"G!Q6FgclF`fm7$Fffm7$$\"3 8wXr)HeME#F0$\"3$>vLC#GfV6Fgcl7$7$F^em$\"3%Ga623'fX6FgclF\\gm7$7$$\"3# *o\"G!Qj0wEF0$\"3Uq-9Zss6%)F*7$$\"3s_T;GYS^CF0$\"3-mVRK()yk%)F*7$F[hm7 $F^em$\"3#*fq^B#Qk^)F*7$7$F^em$\"3tuopl)\\]-\"Fgcl7$$\"3rc(Qc13]Y#F0$ \"3(z<&GXr'y-\"Fgcl7$7$Fggm$\"3Q!\\5](*p)G5FgclFhhm7$Fbgm7$$\"3r8bvux? 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The values returned are: \n1.the sym bolic expressions for the " }{XPPEDIT 18 0 "c[1] " "&%\"cG6#\"\"\"" } {TEXT -1 37 " constant\n2.the dominant singularity " }{XPPEDIT 18 0 "r ho" "I$rhoG6\"" }{TEXT -1 0 "" }{TEXT -1 24 "\n3.the Puiseux expansion " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1012 "singExpansion:=proc(e q, y, z)\nlocal rho_symb, \n deq, tau_vals, \n all_pui s, all_puis_s, c1;\n\n#--numerical and symbolic representation of the \+ singularities\nrho_symb:=locateDominantSing(eq, y, z);\n\n#--values of the function at the singularity\ndeq:=subs(z=rho_symb, diff(eq,y));\n print(`Candidate values for function at singularity`);\ntau_vals:=[fso lve(deq,y)];\nprint(tau_vals);\n\n#--Puiseux expansion; we expect a si ngle exp. to have a t term\nall_puis:=algeqtoseries(subs(z=rho_symb*(1 -t^2),eq),t,y,6);\n\n#--convert into series i.e. remove the O()\nall_p uis_s:=map(eval, map2(subs, O=0, all_puis));\n#--collect the coeff in \+ t and select the non-null one(s)\nc1:=map(proc(x) if x<>0 then x fi en d, map(coeff, all_puis_s, t, 1));\n\nif nops(c1)<>1 then\n print(`Prob lem in singular expansion`); print(all_puis); RETURN(FAIL)\nelse\n c1: =op(c1); if evalf(c1)>0 then c1:=-c1 fi\nfi;\n\n#--returns the constan t c1 and the singularity in symbolic forms\nprint(`c1 constant`, c1,ev alf(c1,20));\n[c1,rho_symb, my_puis]\nend:" }}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 43 "Applications to non-crossing configurations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 170 "We now present the whole chain that goes from the grammars specification to the asymptotic machinery. More pre cisely, for each of the NC configurations studied above, we:" }}{PARA 0 "" 0 "" {TEXT -1 127 "1.call Combstruct[gfeqns] to retrieve the syst em of equations verified by the generating functions associated with t he grammar," }}{PARA 0 "" 0 "" {TEXT -1 98 "2.compute the algebraic eq uation verified by a given generating function with gfun[algfuntoalgeq ]," }}{PARA 0 "" 0 "" {TEXT -1 66 "3.feed this equation to the asympto tic machinery developped above." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 5 "Trees" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Generating functions a ssociated with the grammar:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "gfeqns(tbf, unlabelled, z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&/ -%\"BG6#%\"zG*&-%\"FGF'\"\"#-%\"ZGF'\"\"\"/F**$,&F/F/F%!\"\"F3/F-F(/-% \"TGF'*&F-F/F2F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "treesSy s:=gfsolve(tbf, unlabelled, z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%) treesSysG<&/-%\"FG6#%\"zG-%'RootOfG6#,(%#_ZG!\"\"*&F/\"\"$F*\"\"\"F3F3 F3/-%\"TGF),$*&F*F3,&F0F3*&F+\"\"#F*F3F3F0F0/-%\"BGF)F:/-%\"ZGF)F*" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Algebraic equation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "trees:=algfuntoalgeq(subs(treesSys, T(z)), y(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&treesG,(*&%\"yG\" \"\"%\"zGF(!\"\"*$F)\"\"#F(*$F'\"\"$F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Asymptotic machinery:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "resTrees:=singExpansion(trees, y, z):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/Singularity~isG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"%\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%MCandidate~values~fo r~function~at~singularityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$!+AAA AA!#5$\"+AAAAAF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%,c1~constantG-%'R ootOfG6#,&!\"%\"\"\"*$%#_ZG\"\"#\"$V#$!5cOo\"*>)f+IG\"!#?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Plot of the singularity:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "implicitplot(trees, z=-0.2..0.2, y= -1..1, numpoints=5000);" }}{PARA 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"6#-%'RootOfG6 $,*\"\"%\"\"\"%#_ZG!#K*$F)\"\"#!\")*$F)\"\"$\"\"&$\"+p5&e@\"!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%MCandidate~values~for~function~at~sin gularityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+?q\")o&)!#5$\"+W(>V @\"!\"*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6%%,c1~constantG-%'RootOfG6#, **$%#_ZG\"\"#\"%p8*$-F%6$,*\"\"%\"\"\"F)!#KF(!\")*$F)\"\"$\"\"&$\"+p5& e@\"!#5F*\"%!H&!$G#F1F-\"$\")*$!5(p/4Q%ob=$\\\"!#?" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 16 "Connected graphs" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Generating functions associated with the grammar:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "CGSys:=gfsolve(ar, unlabelle d, z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&CGSysG<%/-%\"ZG6#%\"zGF*/ -%#EAGF)-%'RootOfG6#,*%#_ZG\"\"\"*$F2\"\"#!\"$*$F2\"\"$F5F*!\"\"/-%\"C GF),$*(F*F3,(F3F3F.F6*$F.F5F5F3,*F9F3F.F8F@!\"#F*F3F9F9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Algebraic equation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "CG:=algfuntoalgeq(subs(CGSys,C(z)), y(z));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#CGG,**$%\"yG\"\"$\"\"\"*$F'\"\"#F) *&F'F)%\"zGF)!\"$*$F-F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Asym ptotic machinery:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "resCG: =singExpansion(CG, y, z):" }}{PARA 8 "" 1 "" {TEXT -1 82 "Error, (in l ocateDominantSing) More than one root of modulus < 1: not implemented! 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C6Fdjp-%'COLOURG6&%$RGBG\"\"\"\"\"!F[_w-%+AXESLABELSG6$%\"zG%\"yG" 2 256 252 252 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 3 0 0 0 0 0 0 }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 11 "Dissections" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "dissSys:=gfsolve(dissG, unlabelled, z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(dissSysG<$/-%#DiG6#%\"zG,(#\"\"\"\"\"%F-F*F,*$,(F-F- F*!\"'*$F*\"\"#F-#F-F3#!\"\"F./-%\"ZGF)F*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 47 "diss:=algfuntoalgeq(subs(dissSys,Di(z)), y(z));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dissG,(*$%\"yG\"\"#F(*&,&!\"\"\"\" \"%\"zGF+F,F'F,F,F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "re sDiss:=singExpansion(diss, y, z):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% /Singularity~isG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RootOfG6$,(\"\" \"F'%#_ZG!\"'*$F(\"\"#F'$\"+`(Gdr\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%MCandidate~values~for~function~at~singularityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#$\"+)=K*GH!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 %%,c1~constantG-%'RootOfG6#,(*$%#_ZG\"\"#\"\")-F%6$,(\"\"\"F/F)!\"'F(F /$\"+`(Gdr\"!#5\"\"$!\"\"F/$!5n4SN_x&GHY#!#?" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 118 "It should be noticed that the equation obtained in the paper and stated below in slightly different.But remember that " } {XPPEDIT 18 0 "z^i" ")%\"zG%\"iG" }{TEXT -1 57 " counts here the numbe r of dissections of a polygon with " }{XPPEDIT 18 0 "i+1" ",&%\"iG\"\" \"\"\"\"F$" }{TEXT -1 57 " vertices. 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G9dG92;'F-7$$!3ImmmmmmmTF-$\"3+++++++]iF-7$7$$!3&ommmmmm;%F-Fc_lF`]l7$ Fi^l7$$!3Q3/-^v(QW$F-$!3=#fz*[C71GF-7$7$F`^l$!3wBAAAAAAs#F-$!3YxxxxxxxAF-7$7$ Fjal$!3k:YQ:YQ:@F-F]bl7$Fj`l7$$!3++++++]7GF-$\"3++++++]PfF-7$7$Fjal$\" 3`9YQ:YQleF-Fgbl7$7$Fjal$!3O:YQ:YQ:@F-7$$!3l3EyM/8kBF-$!3N\"R<_cp3,#F- 7$7$$!3YZ*y:j_5<#F-$!3+++++++v=F-Fdcl7$Fjcl7$$!3j>7&>7&>7?F-$!3k!y[!y[ !yt\"F-7$7$F]dl$!3o&p3EyM/j\"F-F`dl7$7$Fjal$\"3k:YQ:YQleF-7$$!3#eJE0@% ot>F-$\"3E:j_5UoBdF-7$7$F]dl$\"3^*************p&F-F]el7$Ffdl7$$!3&ommm mmmm\"F-$!3:LLLLLLe9F-7$7$$!3&*)))))))))))))Q\"F-$!3+++++++]7F-Fgel7$F ]fl7$$!3Vnvcnvc<8F-$!3VKCVKCV#=\"F-7$7$F`fl$!3.++++++D6F-Fcfl7$7$F`fl$ \"3W+++++++bF-7$$!3)p&G9dG92;F-$\"3+++++++DcF-7$F`glFcel7$Fifl7$$!34z: j_5Uo)*!#>$!3\"4Uot%*y:)))Fjgl7$7$$!3\\HN#)eqkw/>w/p8F-7$7$Fd\\m$\"3++++++]i:F-Fg\\m7$7$Fd\\m$\"3++++++]iSF- 7$$\"3QAAAAAAA(*Fjgl$\"3+++++++vVF-7$Fd]mFg[m7$F]]m7$$\"3ajjjjjjj8F-$ \"3YOOOOOOh " 0 "" {MPLTEXT 1 0 40 "partsSys:=gfsolve(partG, unlabelled , z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)partsSysG<%/-%\"ZG6#%\"zGF */-%\"VGF),&#\"\"\"\"\"#F0*$,&F0F0F*!\"%F/#!\"\"F1/-%\"PGF),$*&F*F6,&F 0F0F2F6F0F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "parts:=algfu ntoalgeq(subs(partsSys,P(z)), y(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%&partsG,(\"\"\"F&*&%\"yG\"\"#%\"zGF&F&F(!\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 37 "resParts:=singExpansion(parts, y, z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%/Singularity~isG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%MCandi date~values~for~function~at~singularityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#$\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>Problem~in~si ngular~expansionG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,,\"\"#\"\"\"% \"tG!\"#*$F'F%F%*$F'\"\"$F(-%\"OG6#*$F'#\"\"(F%F&,,F%F&F'F%F)F%F*F%F,F &" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)resPartsG%%FAILG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Here we are in trouble since there are 2 \+ Puiseux expansions with " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 0 " " }{TEXT -1 111 " terms. But only the first one makes sense since the generating function increases with its argument. So that:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "resParts[1]:=2: resParts[2]:=1/4:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Plot of the singularity:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "implicitplot(parts, z=-2..1, y=-5..5, numpoints=1000);" }}{PARA 13 "" 1 "" {INLPLOT "6$-%'CURVESG6 hr7$7$$!\"#\"\"!$!363uS2uS2**!#=7$$!3!eA.Hh^9)>!#<$!3dsC*)p&zK***F-7$7 $$!3++++++D1>F1$!3S(4a9oy,-\"F1F.7$7$F($\"3FI\"R<_cp$[F-7$$!3RRws9j.e> F1$\"3H=8#4\\57&[F-7$7$F6$\"30;tGq6zB\\F-F>7$F57$$!34,&HC3Ur(=F1$!3'Hm ,>REX.\"F17$7$$!3++++++]7=F1$!3()H'eX1*o`5F1FH7$FD7$$!3\"\\ya0gxo'=F1$ \"3s+&f^o'eP\\F-7$7$FO$\"3XhD`_2G;]F-FT7$FN7$$!3yv.%yT(4r2' 3b2\"F17$7$$!3++++++v=UGgi*\\6&F-Fdo7$F^o7$$!3K#4/Cm^Gm\"F1$!3'*fj)> zFQ7\"F17$7$$!3+++++++D;F1$!3'))>0H:]o8\"F1F^p7$7$F_o$\"30=UGgi*\\6&F- 7$$!3_$*=siw2&o\"F1$\"3f\\kRd()eF^F-7$7$Fep$\"3(z6%HN#)e?_F-F]q7$Fdp7$ $!3.Gv(zhbn'4pW`F-7$7$$!3++++++]P9F1$\"3A\"p&e[3\\baF-Fbt7$7 $FitFcs7$$!3DIu([\"G*pV\"F1$!3mK_2<1p^7F17$7$$!3++++++vV8F1$!3P7)GI.6T I\"F1F_u7$Fht7$$!3r4$>X]eRT\"F1$\"3g#*p(R[$GlaF-7$7$$!3y*********\\PM \"F1$\"3wZyc*\\Yme&F-F[v7$Feu7$$!3jvrE:!3yy6gma\"F1Fby7$F^y7$$!3f`5BvtOX6F1$\"3G?^.TeC()eF-7$7$Fiy$ \"3'yy)F-$\"3Ex#eXQ1iP'F-7$7$Fh\\l$\"3-SESESE!R'F-Fi]l7$7$$!3++ ++++]7yF-$!3Y;bAtX2M>F17$$!3Eyxxxxxx\")F-$!3+++++++v=F17$7$$!3:xxxxxxx \")F-F[_lFg\\l7$7$Fh\\l$\"38TESESE!R'F-7$$!3%)\\etPg\"o*yF-$\"3'o\\>\" fM0JlF-7$7$Fd^l$\"3&HM.E#RUklF-Fe_l7$7$Fd^l$!3o;bAtX2M>F17$$!3\"GguJgu 1H(F-$!31*zT*)zT*[?F17$7$$!3+++++++voF-$!3g\"o$4BmWI@F1Fb`l7$F[`l7$$!3 \\Tnpy9f6qF-$\"3mQ\"*)*G\\I0nF-7$7$Fi`l$\"3gA'*>uE=jnF-F^al7$7$$!3++++ ++]PfF-$!3/!*Hz_'3xesF-F]dl7$7$$!3++++++]iSF-$!3Jr86&)em>KF17$$!3)************RA% F-$!3+++++++DJF17$F\\el7$$!3T([!y[!y[b%F-$!3YP)RP)R(3'HF17$7$$!3(Hv>k3 `(>[F-$!3++++++]7GF1Fbel7$FhelFacl7$Fcdl7$$!3,@Vl()4K/WF-$\"3Gp5=#H.%* Q(F-7$7$Fhdl$\"3+/ks;!yJd(F-F_fl7$7$$F`elF-$!3PhY7mCh'*RF17$$!3)yxxxxx xP$F-$!3+++++++]PF17$F]gl7$$!3ixz$*[7*))e$F-$!3]Sto$ep`f$F17$7$$!3MYk# 3!>PbPF-$!3++++++]PMF1Fcgl7$Figl7$Fhdl$!3vr86&)em>KF17$Fefl7$$!3_wH*yO 5\"eNF-$\"3P@*4jA,Pp(F-7$7$Fjfl$\"3%yz;c1v%\\zF-Fchl7$7$$!3\"********* *****R#F-$!\"&F*7$$!3*>=Xnb?(yCF-$!3yR\\3\"[EH!\\F17$7$$!3LWWWWWW)e#F- $!3++++++](o%F1Fbil7$Fhil7$$!3*3?IsunFu#F-$!3/mK#4v5C]%F17$7$$!3EAhIlK ;3GF-$!3+++++++vVF1F^jl7$Fdjl7$$!3,$4-y!=gbIF-$!3mNEtIFj&3%F17$7$$!3jv F-$\"3k#)*yI54c[)F-7$7$$F]tF-$\"3rRY$*p\"e(y*)F-Fd\\m7$Fj\\m7$$!3Sr&G 92T<9\"F-$\"3m/>w/p89!*F-7$7$$!356666666r!#>$\"3+++++++v$*F-F_]m7$Fe]m 7$$!3W>/Truw3QFh]m$\"3YS,Z!\\AHg*F-7$7$$F`elFh]m$\"3^QQqX]#Hq*F-F\\^m7 $Fb^m7$$\"3xvBfy_?0OFh]m$\"3!3#p/C)fc-\"F17$7$$Fg]lFh]m$\"321Zw6%HN2\" F1Fg^m7$F]_m7$$\"3[0(4s#faI5F-$\"3NJMEC!=[6\"F17$7$$\"3s**********\\i: F-$\"3\\k'[G#z4T7F1Fb_m7$7$$\"3++++++]i:F-$\"3rk'[G#z4T7F17$$\"3hp,h'z n:e\"F-$\"3I5mznSkV7F17$7$$\"3.+++++++;F-$\"3+++++++]7F1Fc`m7$Fi`m7$$ \"3#zZCt#G8!4#F-$\"3ZS=*3RAmQ\"F17$7$$\"3%************RI#F-$F``mF1F_am 7$Feam7$$\"3g8Vyg>!*zBF-$\"3#F17$7$$\"3\"**************R#F-$\"3++++++++DF1Fec m7$F[dm7$$\"3o#R(eb%4$3CF-$\"3F-U![^j0`#F17$7$$\"3&yN\"pC!e8H#F-$\"3++ ++++]7GF1Fadm7$Fgdm7$$\"3(=Mh;'R_(H#F-$\"3j_&zF,#**zGF17$7$$\"3;++++++ w@F-$\"3+++++++DJF1F]em7$Fcem7$$\"3q:TC^u;t@F-$\"3&zi=H=WRB$F17$7$$\"3 %=Pbt\"*4G1#F-$\"3++++++]PMF1Fiem7$F_fm7$$\"3H!*e4T!*3Z?F-$\"3v-Zj>.Z) e$F17$7$$\"3Qbbbbbbb>F-$\"3+++++++]PF1Fefm7$7$$\"3mbbbbbbb>F-F^gm7$$\" 3[x'zFt+b#>F-$\"3D2,ub(*\\TRF17$7$$\"3sZ^x,8ib=F-$\"3++++++]iSF1Fdgm7$ Fjgm7$$\"3YV%43kS8\"=F-$\"3)e=I(>J0#H%F17$7$$\"3.\\C71`Ej " 0 " " {MPLTEXT 1 0 65 "nbEnt:=proc(c1,rho, n)\n evalf(c1*rho^(-n)/(2*sqrt( Pi*n^3))) \nend;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&nbEntG:6%%#c1G% $rhoG%\"nG6\"F*F*-%&evalfG6#,$*(9$\"\"\")9%,$9&!\"\"F1-%%sqrtG6#*&%#Pi GF1F5\"\"$F6#F1\"\"#F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "Comp arisons with the exact values are as follows. It is interesting to obs erve that these values are within about 2% in any case:" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 5 "Trees" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "exactTr:=evalf(count([T,tbf], size=100));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(exactTrG$\"5zCJx*[UH76$\"#f" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "estimTr:=nbEnt(resTrees[1],resTrees [2],100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(estimTrG$\"5xtK\\\\TB= $3$\"#f" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "exactTr/estimTr; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5uX()p%oz'445!#>" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "Forests" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "exactFo:=evalf(count([F,fo], size=100));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(exactFoG$\"5gkKDp;fP%Q\"\"#o" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "estimFo:=nbEnt(resForests[1], resFo rests[2], 100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(estimFoG$\"5RMoR + " 0 "" {MPLTEXT 1 0 16 "exactFo/estim Fo;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5M7\\#)=iZ265!#>" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 16 "Connected graphs" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "exactCo:=evalf(count([C,ar],size=100));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(exactCoG$\"5I&QK+3d$****[\"#x" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "estimCo:=nbEnt(resCG[1],resC G[2],100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(estimCoG$\"5Wl*>&)fjX V#[\"#x" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "exactCo/estimCo; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5WOGV.b/o:5!#>" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 14 "General graphs" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "exactGG:=evalf(count([G, br], size=100));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(exactGGG$\"5(RU6'ew#)\\Fb\"##)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "estimGG:=nbEnt(resGG[1], res GG[2], 100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(estimGGG$\"5!*)R7)* Q#[SDa\"##)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "exactGG/esti mGG;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5\"e_k4am<)=5!#>" }}}} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 11 "Dissections" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 47 "exactDiss:=evalf(count([Di, dissG], size=100)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*exactDissG$\"57?I#ocbFL]#\"#` " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "estimDiss:=nbEnt(resDis s[1], resDiss[2], 100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*estimDis sG$\"5O4&QVrb-X\\#\"#`" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "e xactDiss/estimDiss;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5;i&)[!*yx`. 5!#>" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 11 "Partitions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "exactPart:=evalf(count([P, partG], \+ size=100));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*exactPartG$\"5p'\\J, 4Z*>l*)\"#P" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "estimPart:=n bEnt(resParts[1], resParts[2], 100);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%*estimPartG$\"5-%oDv(fq " 0 "" {MPLTEXT 1 0 20 "exactPart/estimPart;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5iYuV\"[k@')))*!#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "References" }}}{PARA 0 "" 0 "" {TEXT -1 98 "[Dr97] M. Drmota, S ystems of functional equations, Random Structures and Algorithms 10, 1 -2, 1997." }}{PARA 0 "" 0 "" {TEXT -1 131 "[FlaNo97] P. Flajolet and M . Noy, Analytic Combinatorics of Non-Crossing Configurations, Rapport \+ de Recherche INRIA No. 3196, 1997." }}{PARA 0 "" 0 "" {TEXT -1 80 "[Ha Pa73] F. Harary and E.M. Palmer, Graphical Enumeration, Academic Press , 1973." }}{PARA 0 "" 0 "" {TEXT -1 100 "[Sloa95] N.J.A. Sloane and S. Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995. " }}}{MARK "6 1 1 7 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }