(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 2171974, 56251]*) (*NotebookOutlinePosition[ 2173006, 56288]*) (* CellTagsIndexPosition[ 2172926, 56282]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ "A ", StyleBox["Mathematica", FontSlant->"Italic"], " Package for Studying Posets " }], "Title", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "Curtis Greene\nEugenie Hunsicker\nDepartment of Mathematics\nHaverford \ College\nJuly 1990\n\n", StyleBox["Updates", FontSlant->"Italic"], ": John Dollhopf, Sam Hsiao, Curtis Greene\n1992-94" }], "Text", TextAlignment->Center, AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[" 1. Introduction", "Section", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ " This document illustrates a package designed to generate, display, \ and explore partially ordered sets (also known as \"posets\"), with an \ emphasis on topics of current interest in combinatorics. The package has \ two distinctive features: (1) a large repertoire of standard examples \ (Boolean lattices, subword posets, lattices of partitions, distributive \ lattices, Young's lattice, Bruhat orders, etc.), and (2) the ability to \ generate a poset directly from its formal definition, i.e., from a ", StyleBox["Mathematica", FontSlant->"Italic"], " program giving its \"covering function\". The latter approach permits a \ compact definition of posets whose structure is extremely complex, or \ unknown. Posets of 100-200 elements can typically be generated and displayed \ within a couple of minutes. Smaller posets can be generated much more \ quickly.\n \n A poset may also be defined by listing its covering \ pairs, or any generating set of ordered pairs, or by giving the incidence \ matrix for such a relation. Also included are some standard constructions of \ new posets from old: product and sum, subposet, distributive lattice of order \ ideals, etc. \n \n Once a poset has been generated, a large \ number of tools are available for investigating its combinatorial structure. \ In the present version of the package the selection of tools implemented \ reflects the authors' interests: enumerative invariants, chains, \ multichains, antichains, rank generating functions, Mobius functions, linear \ extensions, and related topics. A good reference for the underlying \ combinatorics is Richard Stanley's book, ", StyleBox["Enumerative Combinatorics", FontSlant->"Italic"], " (Wadsworth, 1986). \n\n Our intention here is not to write a \ \"users manual\", but rather to give an informal demonstration of the \ package's principal features. For a complete list of commands and syntax, \ the user is advised to print out the \"usage\" section of the package, which \ has been written with this in mind.\n \n The package is available \ on the web at ", StyleBox["http://www.haverford.edu/math/cgreene.html", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], ". It is available in both notebook and plain ascii text formats. The \ authors welcome comments, questions, bug reports, criticisms, and other \ suggestions. Such communications should be directed to ", StyleBox["cgreene@haverford.edu", FontWeight->"Bold"], ", or to Curtis Greene, Department of Mathematics, Haverford College, \ Haverford PA 19041. " }], "Text", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}] }, Closed]], Cell[CellGroupData[{ Cell[" 2. Using the Package", "Section", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "The current version of the package (July 1994) is called ", StyleBox["Posets22d.nb.", FontWeight->"Bold"], " For convenience it's name can be changed, for example, to ", StyleBox["Posets.nb", FontWeight->"Bold"], ". It should be placed in a directory or folder on ", StyleBox["Mathematica", FontSlant->"Italic"], "'s search path for input. For example, Macintosh users should place it in \ either the ", StyleBox["Mathematica", FontWeight->"Bold"], " or ", StyleBox["Packages", FontWeight->"Bold"], " folder. Or you can give the complete path specification. To load the \ package, type (for example)" }], "Text", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(<< "\"\)], "Input", InitializationCell->True, AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell["To generate a poset, type", "Text", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell["Build[,]", "Input", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "where ", StyleBox["", FontWeight->"Bold"], " is a \"definition\" (of one of three types, to be described below), and \ ", StyleBox["", FontWeight->"Bold"], " is a string used to identify the poset in subsequent computations. If no \ name is supplied, the package creates one automatically. " }], "Text", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "The elements of a poset may be represented by any object recognized by ", StyleBox["Mathematica", FontSlant->"Italic"], ", typically by lists, strings, or integers. These objects become \ \"labels\" for elements of the poset. A poset definition may have any one of \ the following forms: " }], "Text", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "(1) A triple ", StyleBox["{f, , H}", FontWeight->"Bold"], " , where ", StyleBox["f", FontWeight->"Bold"], " is a ", StyleBox["Mathematica", FontSlant->"Italic"], " function that generates, from an element ", StyleBox["w", FontWeight->"Bold"], " of P a list consisting of \n its immediate successors (or covers); \ ", StyleBox["", FontWeight->"Bold"], " is an explicit list of of the minimal elements of P; and ", StyleBox["H", FontWeight->"Bold"], " is an upper bound\n for the height of the poset (not necessarily the \ exact height.)\n\n(2) A pair ", StyleBox["{, N}", FontWeight->"Bold"], " , where ", StyleBox["N", FontWeight->"Bold"], " is an integer and ", StyleBox["", FontWeight->"Bold"], " is a set of pairs of integers \[LessEqual] N. This should be an \n \ acyclic relation generating the partial order. It need not be a transitive \ relation, nor must it be a minimal generating set.\n\n(3) A matrix ", StyleBox["M", FontWeight->"Bold"], ", which is the incidence matrix of an acyclic relation generating the \ partial order. Again, the relation represented by \n ", StyleBox["M", FontWeight->"Bold"], " need not be transitive, nor must it be minimal." }], "SmallText", CellFrame->True, AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell["The effect of ", "Text", PageBreakAbove->False, AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell["Build[,]", "Input", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ is to generate three fundamental objects associated with the poset:\ \>", "Text", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ StyleBox["(1) An object ", FontSize->10], StyleBox["P[]", FontSize->10, FontWeight->"Bold"], StyleBox[". This is a list of the elements of P. The elements of ", FontSize->10], StyleBox["P[]", FontSize->10, FontWeight->"Bold"], StyleBox[" serve as \"labels\" for P. If P is \n defined by order \ relations or by an incidence matrix, then the labels are just indices, i.e. \ ", FontSize->10], StyleBox["P[]", FontSize->10, FontWeight->"Bold"], StyleBox[" = \n ", FontSize->10], StyleBox["{1,2, ..., Card[P]}. \n ", FontSize->10, FontWeight->"Bold"], StyleBox["\n(2) An object ", FontSize->10], StyleBox["Rank[]", FontSize->10, FontWeight->"Bold"], StyleBox[". This is a list showing the rank of each element of", FontSize->10], StyleBox[" ", FontSize->10, FontWeight->"Bold"], StyleBox["P. The rank of an element ", FontSize->10], StyleBox["x", FontSize->10, FontWeight->"Bold"], StyleBox[" is defined as the\n length of the longest chain descending \ from ", FontSize->10], StyleBox["x", FontSize->10, FontWeight->"Bold"], StyleBox[". The indexing of ", FontSize->10], StyleBox["Rank[]", FontSize->10, FontWeight->"Bold"], StyleBox[" agrees with that of ", FontSize->10], StyleBox["P[]. ", FontSize->10, FontWeight->"Bold"], StyleBox["\n \n(3) An object ", FontSize->10], StyleBox["CoverRelations[]", FontSize->10, FontWeight->"Bold"], StyleBox[". This is a list of pairs of indices ", FontSize->10], StyleBox["{i,j}", FontSize->10, FontWeight->"Bold"], StyleBox[" that defines a minimal generating set for the \n order \ relation.", FontSize->10] }], "Text", CellFrame->True, AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ All of the other commands and operations in the package refer to these \ objects. Many other functions and derived objects are available. For a full \ list, with precise syntax, refer to the \"usage\" portion of the package \ itself. An index of all commands may be found at the end of this document. \ \ \>", "Text", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "In cases where functions depend on results returned by other functions, \ these are called automatically. For example, many functions need to compute \ ", StyleBox["ZetaP[]", FontWeight->"Bold"], ", the incidence matrix of the transitive closure of the cover relation, \ and this is done automatically when required. The user need not be concerned \ with the order in which comands are executed. Once computed, objects are \ \"remembered\", so that time is not wasted computing them again.\n\nElements \ of the poset are always maintained in a \"natural\", or topologically sorted \ order. When P is a ranked (or graded) poset, the elements of each rank are \ grouped together, and a separate object ", StyleBox["PGraded[][[k]]", FontWeight->"Bold"], " lists the elements of rank k. " }], "Text", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ By default, messages are produced to indicate the status of computations in \ progress. If you most of these messages to be suppressed, type\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(\(Verbose = False;\)\)], "Input", InitializationCell->True, AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell["This notebook was executed with ", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(Version\)], "Input", InitializationCell->True, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \("2.2d4b"\)], "Output"] }, Open ]], Cell["of the package.", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ StyleBox["Note to Macintosh users.", FontWeight->"Bold"], " Because of the large number of graphic displays, it is difficult to \ execute this documentation notebook its entirety on a small Macintosh system. \ We recommend using virtual memory with at least 3MB devoted to the front end, \ and at least 6MB devoted to the kernel. If this is impractical or \ inconvenient, the notebook can easily be read easily in several passes, \ restarting in between. This advice applies only to the documentation; for \ less elaborate tasks the standard 1.5MB/5MB partition is usually adequate to \ run the package, without virtual memory." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ StyleBox["Note to users of other systems.", FontWeight->"Bold"], " All of the commands in this document may be executed using ordinary \ XWindows versions of ", StyleBox["Mathematica", FontSlant->"Italic"], " (e.g., on Suns) provided the appropriate graphics windows have been \ initialized. Just load the plain ascii version of the package and enter the \ commands manually. An abbreviated version of this text has been included \ with the distribution files, but it does not function as a \"notebook\". The \ notebook itself should run on other systems that support ", StyleBox["Mathematica", FontSlant->"Italic"], " with notebooks, but we have not tested it on these platforms. " }], "Text", ImageRegion->{{0, 1}, {0, 1}}] }, Closed]], Cell[CellGroupData[{ Cell[" 3. Fundamental Families of Posets", "Section", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[" SUBSETS OF A SET", "Subsubsection", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "In the first example we generate the lattice of subsets of a 4-element \ set, and several combinatorial invariants. Remember that ", StyleBox["Subsets[4]", FontWeight->"Bold"], " is the built-in \"definition\" of the poset, and ", StyleBox["sub4", FontWeight->"Bold"], " is a \"tag name\" (chosen by the user) to reference all of the objects \ associated with this poset." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(Build[Subsets[4], sub4]\)], "Input", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ InterpretationBox[\("Building poset "\[InvisibleSpace]sub4\ \[InvisibleSpace]" ..."\), SequenceForm[ "Building poset ", sub4, " ..."], Editable->False]], "Print"], Cell[BoxData[ \("Done"\)], "Print"] }, Open ]], Cell["\<\ Once a poset has been \"built\", its Hasse diagram may be displayed:\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(Diagram[sub4]\)], "Input", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 0.103006 [ [ 0 0 -0.125 0 ] [ 0 0 -0.125 0 ] [ 0 .61803 .125 0 ] [ 1 0 .125 0 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash 0 0 m 1 0 L s 0 0 m 0 .61803 L s 0 .61803 m 1 .61803 L s 1 0 m 1 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .0028 w .5 .10301 m .2 .20601 L s .5 .10301 m .4 .20601 L s .5 .10301 m .6 .20601 L s .5 .10301 m .8 .20601 L s .2 .20601 m .14286 .30902 L s .2 .20601 m .28571 .30902 L s .2 .20601 m .42857 .30902 L s .4 .20601 m .14286 .30902 L s .4 .20601 m .57143 .30902 L s .4 .20601 m .71429 .30902 L s .6 .20601 m .28571 .30902 L s .6 .20601 m .57143 .30902 L s .6 .20601 m .85714 .30902 L s .8 .20601 m .42857 .30902 L s .8 .20601 m .71429 .30902 L s .8 .20601 m .85714 .30902 L s .14286 .30902 m .2 .41202 L s .14286 .30902 m .4 .41202 L s .28571 .30902 m .2 .41202 L s .28571 .30902 m .6 .41202 L s .42857 .30902 m .4 .41202 L s .42857 .30902 m .6 .41202 L s .57143 .30902 m .2 .41202 L s .57143 .30902 m .8 .41202 L s .71429 .30902 m .4 .41202 L s .71429 .30902 m .8 .41202 L s .85714 .30902 m .6 .41202 L s .85714 .30902 m .8 .41202 L s .2 .41202 m .5 .51503 L s .4 .41202 m .5 .51503 L s .6 .41202 m .5 .51503 L s .8 .41202 m .5 .51503 L s .02 w .5 .10301 Mdot .2 .20601 Mdot .4 .20601 Mdot .6 .20601 Mdot .8 .20601 Mdot .14286 .30902 Mdot .28571 .30902 Mdot .42857 .30902 Mdot .57143 .30902 Mdot .71429 .30902 Mdot .85714 .30902 Mdot .2 .41202 Mdot .4 .41202 Mdot .6 .41202 Mdot .8 .41202 Mdot .5 .51503 Mdot % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{281.813, 174.188}, ImageMargins->{{34, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgoooo0`0002Goool00`000?ooooooo`0Soooo00<0003oooooool08oooo`<0001@oooo0@00 000000<0003oooooool0Boooo`<0000Woooo00<0003oooooool09Oooo`030000oooooooo02Goool3 0000COooo`40000000030000oooooooo04Soool30000:Oooo`030000oooooooo02Ooool00`000?oo ooooo`0Woooo0`0004[oool1000000000`000?ooooooo`15oooo0`0002_oool00`000?ooooooo`0Y oooo00<0003oooooool0:Oooo`<00017oooo0@00000000<0003oooooool0@oooo`80000]oooo00<0 003oooooool0:oooo`030000oooooooo02_oool20000AOooo`40000000030000oooooooo043oool3 0000;_ooo`030000oooooooo02goool00`000?ooooooo`0/oooo0`0004;oool1000000000`000?oo ooooo`0moooo0`00033oool00`000?ooooooo`0_oooo00<0003oooooool0;_ooo`<0000ooooo0@00 000000<0003oooooool0=?ooo`<00003oooo0`0002ooool40000oooo0P0000?oool300007Oooo`<00009oooo00<0003oooooool01oooo`<0000L oooo100000Woool00`000?ooooooo`02oooo00<0003oooooool0 oooo00<0003oooooool03Oooo`<0000@oooo100000koool00`000?ooooooo`05oooo00<0003ooooo ool0<_ooo`40000000030000oooooooo037oool00`000?ooooooo`07oooo00<0003oooooool01_oo o`80000Eoooo0P0001_oool200001oooo`<0000;oooo0`00017oool00`000?ooooooo`0@oooo0P00 00_oool300004Oooo`030000oooooooo00Ooool00`000?ooooooo`0aoooo0@00000000<0003ooooo ool0oooo0`00 00koool00`000?ooooooo`0>oooo00<0003oooooool0;_ooo`40000000030000oooooooo02koool0 0`000?ooooooo`0?oooo00<0003oooooool02Oooo`800003oooo0P0002[oool300000_ooo`80000? 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For the subset lattice, these are just binomial \ coeffients.\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(RGF[sub4]\)], "Input", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(1 + 4\ q + 6\ q\^2 + 4\ q\^3 + q\^4\)], "Output"] }, Open ]], Cell[TextData[{ " ", StyleBox["P[sub4]", FontWeight->"Bold"], " is one of the three fundamental objects produced by the ", StyleBox["Build", FontWeight->"Bold"], " command. It contains the labels of all of the elements just generated." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(P[sub4]\)], "Input", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \({{1, 2, 3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2}, {1, 3}, {2, 3}, {1, 4}, {2, 4}, {3, 4}, {1}, {2}, {3}, {4}, {}}\)], "Output"] }, Open ]], Cell[TextData[{ "A second fundamental object is the ", StyleBox["Rank[sub4]", FontWeight->"Bold"], ", which shows the rank of each element." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(Rank[sub4]\)], "Input", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \({0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4}\)], "Output"] }, Open ]], Cell["\<\ We can inquire whether this poset is a lattice. The answer is, of course, \ \"yes\".\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(LatticeQ[sub4]\)], "Input", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[{ "The \"Zeta Polynomial\" Z[n] of a poset P counts the number of multichains \ x", StyleBox["1", FontVariations->{"CompatibilityType"->"Subscript"}], "\[LessEqual]x", StyleBox["2", FontVariations->{"CompatibilityType"->"Subscript"}], " ... \[LessEqual] x", StyleBox["n-1", FontVariations->{"CompatibilityType"->"Subscript"}], " that can be formed from the elements of P. " }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(ZetaPoly[sub4, n]\)], "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(n\^4\)], "Output"] }, Open ]], Cell[TextData[{ "When P is an interval [a,b] this is the same as the number of multichains \ \n a = x", StyleBox["0 ", FontVariations->{"CompatibilityType"->"Subscript"}], "\[LessEqual] x", StyleBox["1 ", FontVariations->{"CompatibilityType"->"Subscript"}], "\[LessEqual] x", StyleBox["2 ", FontVariations->{"CompatibilityType"->"Subscript"}], "\[LessEqual] ... \[LessEqual] x", StyleBox["n", FontVariations->{"CompatibilityType"->"Subscript"}], " = b \nbetwen a and b. The command ", StyleBox["ZetaPoly[name,a,b,n]", FontWeight->"Bold"], " computes the Zeta polynomial of the interval [a,b] in P." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(Table[ZetaPoly[sub4, 1, b, n], {b, 1, Card[sub4]}]\)], "Input", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \({1, n, n, n, n, n\^2, n\^2, n\^2, n\^2, n\^2, n\^2, n\^3, n\^3, n\^3, n\^3, n\^4}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[" PERMUTATIONS: WEAK ORDER ", "Subsubsection", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "The ", StyleBox["weak order", FontSlant->"Italic"], " on permutations is defined by saying that ", StyleBox["w", FontWeight->"Bold"], " covers ", StyleBox["u ", FontWeight->"Bold"], "if", StyleBox[" w ", FontWeight->"Bold"], "can be obtained from", StyleBox[" u ", FontWeight->"Bold"], "by transposing a pair of adjacent increasing elements.", StyleBox[" ", FontWeight->"Bold"] }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(Build[WeakS[4], weaks4]\)], "Input", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ InterpretationBox[\("Building poset "\[InvisibleSpace]weaks4\ \[InvisibleSpace]" ..."\), SequenceForm[ "Building poset ", weaks4, " ..."], Editable->False]], "Print"], Cell[BoxData[ \("Done"\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(P[weaks4]\)], "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \({{1, 2, 3, 4}, {2, 1, 3, 4}, {1, 3, 2, 4}, {1, 2, 4, 3}, {2, 3, 1, 4}, {2, 1, 4, 3}, {3, 1, 2, 4}, {1, 3, 4, 2}, {1, 4, 2, 3}, {3, 2, 1, 4}, {2, 3, 4, 1}, {2, 4, 1, 3}, {3, 1, 4, 2}, {1, 4, 3, 2}, {4, 1, 2, 3}, {3, 2, 4, 1}, {2, 4, 3, 1}, {4, 2, 1, 3}, {3, 4, 1, 2}, {4, 1, 3, 2}, {3, 4, 2, 1}, {4, 2, 3, 1}, {4, 3, 1, 2}, {4, 3, 2, 1}}\)], "Output"] }, Open ]], Cell[TextData[{ "The Hasse diagram for this poset may also be regarded as Cayley graph for \ the symmetric group S", StyleBox["4", FontVariations->{"CompatibilityType"->"Subscript"}], ", with respect to the generators (1,2), (2,3), (3,4). 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oaKoool1000000000`000?ooooooo`3ooooo5_ooo`40000000030000oooooooo0?oooolFoooo0@00 000000<0003oooooool0oooooaKoool1000000000`000?ooooooo`3ooooo5_ooo`40000000030000 oooooooo0?oooolFoooo0@00000000<0003oooooool0oooooaKoool1000000000`000?ooooooo`3o oooo5_ooo`4000000?l0000J00000000\ \>"], ImageRangeCache->{{{0, 280.813}, {173.188, 0}} -> {-0.00105315, \ -4.00002*^-05, 0.0035686, 0.0461929}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ The rank generating function for the weak order factors nicely:\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[{ \(RGF[weaks4]\), "\n", \(Factor[RGF[weaks4]]\)}], "Input", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(1 + 3\ q + 5\ q\^2 + 6\ q\^3 + 5\ q\^4 + 3\ q\^5 + q\^6\)], "Output"], Cell[BoxData[ \(\((1 + q)\)\^2\ \((1 + q\^2)\)\ \((1 + q + q\^2)\)\)], "Output"] }, Open ]], Cell[TextData[{ "The elements at each rank are stored and displayed in the order generated \ by the ", StyleBox["Build", FontWeight->"Bold"], " command. The poset can look quite different if another order is used. \ For example, the next command puts each rank into lexicographic order." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(SortByRanks[weaks4]\)], "Input", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "When elements are represented by lists (as in this case), the labels are \ sometimes hard to read. ", StyleBox["Compact", FontWeight->"Bold"], " command makes lists into strings, and ", StyleBox["Relabel", FontWeight->"Bold"], " replaces each label in ", StyleBox["P[weaks4]", FontWeight->"Bold"], " by its \"compact\" equivalent. 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