<?xml version="1.0" encoding="UTF-8"?>
<Worksheet>
<Version major="2023" minor="0"/>
<Label-Scheme value="2" prefix=""/>
<View-Properties presentation="false" autoexpanding_sections="true" UserProfileName="Maple Default Profile" NumericFormat-ApplyInteger="true" NumericFormat-ApplyRational="true" NumericFormat-ApplyExponent="false" editable="true">
<Hide name="Group Range"/>
</View-Properties>
<MapleNet-Properties prettyprint="3" warnlevel="3" compactdisplay="false" preplot="" helpbrowser="standard" displayprecision="-1" echo="1" scientificx="true" unitattributes="&quot;fontweight&quot; = &quot;bold&quot;" imaginaryunit="I" longdelim="true" elisiontermsthreshold="10000" elisiondigitsafter="100" elisiondigitsbefore="100" plotdevice="inline" errorbreak="1" plotoptions="" plotdriver="opengl" quiet="false" elisiontermsbefore="100" elisiontermsafter="100" historytimestamp="false" screenwidth="79" indentamount="4" plotoutput="terminal" screenpixelheight="1080" rtablesize="[10, 10]" useclientjvm="true" format="worksheet" labelwidth="20" postplot="" typesetting="extended" ansi="false" ansicolor="[]" elisiondigitsthreshold="10000" showassumed="1" ansilprint="false" trailingsemicolon="true" errorcursor="false" labelling="true" screenheight="25" prompt="&gt; " verboseproc="1" latexwidth="8.0" ShowLabels="false"/>
<Styles>
<Font name="Heading 1" background="[255,255,255]" bold="true" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="18" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Warning" background="[255,255,255]" bold="false" executable="false" family="Courier New" foreground="[0,0,255]" italic="false" opaque="false" readonly="true" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="2D Output" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,255]" italic="false" opaque="false" readonly="true" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Heading 4" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="true" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Line Printed Output" background="[255,255,255]" bold="false" executable="false" family="Courier New" foreground="[0,0,255]" italic="false" opaque="false" readonly="true" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Heading 2" background="[255,255,255]" bold="true" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="16" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Maple Output" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="2D Inert Output" background="[255,255,255]" bold="false" executable="true" family="Times New Roman" foreground="[144,144,144]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Heading 3" background="[255,255,255]" bold="true" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="true" opaque="false" readonly="false" size="14" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Diagnostic" background="[255,255,255]" bold="false" executable="false" family="Courier New" foreground="[40,120,40]" italic="false" opaque="false" readonly="true" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Ordered List 1" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Maple Input" background="[255,255,255]" bold="true" executable="true" family="Courier New" foreground="[120,0,14]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Text Output" background="[255,255,255]" bold="false" executable="false" family="Courier New" foreground="[0,0,255]" italic="false" opaque="false" readonly="true" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Ordered List 2" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Ordered List 3" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Ordered List 4" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Ordered List 5" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Annotation Title" background="[255,255,255]" bold="true" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="18" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Header and Footer" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="10" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="HyperlinkError" background="[255,255,255]" bold="false" executable="false" family="Courier New" foreground="[255,0,255]" italic="false" opaque="false" readonly="true" size="12" subscript="false" superscript="false" underline="true" placeholder="false"/>
<Font name="Atomic Variable" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[175,0,175]" italic="true" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="HyperlinkWarning" background="[255,255,255]" bold="false" executable="false" family="Courier New" foreground="[0,0,255]" italic="false" opaque="false" readonly="true" size="12" subscript="false" superscript="false" underline="true" placeholder="false"/>
<Font name="Dictionary Hyperlink" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[147,0,15]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="true" placeholder="false"/>
<Font name="2D Math" background="[255,255,255]" bold="false" executable="true" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Bullet Item" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Maple Plot" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Annotation Text" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="List Item" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Dash Item" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="2D Input" background="[255,255,255]" bold="false" executable="true" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Error" background="[255,255,255]" bold="false" executable="false" family="Courier New" foreground="[255,0,255]" italic="false" opaque="false" readonly="true" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Title" background="[255,255,255]" bold="true" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="18" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Text" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Normal" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Caption Reference" background="[255,255,255]" bold="true" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Code" background="[255,255,255]" bold="false" executable="false" family="Courier New" foreground="[255,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Maple Input Placeholder" background="[255,255,255]" bold="true" executable="true" family="Courier New" foreground="[200,0,200]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="true"/>
<Font name="Equation Label" background="[255,255,255]" bold="true" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Author" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Font name="Hyperlink" background="[255,255,255]" bold="false" executable="false" family="Times New Roman" foreground="[0,128,128]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="true" placeholder="false"/>
<Font name="Caption Text" background="[255,255,255]" bold="true" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" opaque="false" readonly="false" size="12" subscript="false" superscript="false" underline="false" placeholder="false"/>
<Layout name="Heading 1" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="8" spacebelow="4" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Warning" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Heading 4" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Line Printed Output" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="any" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Heading 2" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="8" spacebelow="2" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Maple Output" alignment="centred" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.3" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Heading 3" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Diagnostic" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="any" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Ordered List 1" alignment="left" bullet="numeric" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="-1" bulletsuffix="."/>
<Layout name="Text Output" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="newline" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Ordered List 2" alignment="left" bullet="alphabetic" firstindent="0" leftmargin="36" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="-1" bulletsuffix="."/>
<Layout name="Ordered List 3" alignment="left" bullet="roman" firstindent="0" leftmargin="72" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="-1" bulletsuffix="."/>
<Layout name="Ordered List 4" alignment="left" bullet="ALPHABETIC" firstindent="0" leftmargin="108" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="-1" bulletsuffix="."/>
<Layout name="Ordered List 5" alignment="left" bullet="ROMAN" firstindent="0" leftmargin="144" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="-1" bulletsuffix="."/>
<Layout name="Annotation Title" alignment="centred" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="12" spacebelow="12" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="HyperlinkError" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="HyperlinkWarning" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Bullet Item" alignment="left" bullet="dot" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Maple Plot" alignment="centred" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="List Item" alignment="left" bullet="indent" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Dash Item" alignment="left" bullet="dash" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="3" spacebelow="3" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Error" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Title" alignment="centred" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="12" spacebelow="12" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Normal" alignment="left" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="0" spacebelow="0" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Layout name="Author" alignment="centred" bullet="none" firstindent="0" leftmargin="0" rightmargin="0" linespacing="0.0" spaceabove="8" spacebelow="8" linebreak="space" pagebreak-before="false" initial="0" bulletsuffix=""/>
<Pencil-style name="Pencil 1" pen-color="[0,0,0]" pen-height="1.0" pen-width="1.0" pen-opacity="1.0"/>
<Pencil-style name="Pencil 2" pen-color="[0,0,255]" pen-height="1.0" pen-width="1.0" pen-opacity="1.0"/>
<Pencil-style name="Pencil 3" pen-color="[0,0,0]" pen-height="3.0" pen-width="3.0" pen-opacity="1.0"/>
<Pencil-style name="Pencil 4" pen-color="[0,0,255]" pen-height="3.0" pen-width="3.0" pen-opacity="1.0"/>
<Pencil-style name="Pencil 5" pen-color="[255,0,0]" pen-height="5.0" pen-width="5.0" pen-opacity="1.0"/>
<Highlighter-style name="Highlighter 5" pen-color="[255,255,0]" pen-height="48.0" pen-width="48.0" pen-opacity="0.8"/>
<Highlighter-style name="Highlighter 3" pen-color="[51,255,0]" pen-height="24.0" pen-width="24.0" pen-opacity="0.8"/>
<Highlighter-style name="Highlighter 4" pen-color="[0,255,255]" pen-height="32.0" pen-width="32.0" pen-opacity="0.8"/>
<Highlighter-style name="Highlighter 1" pen-color="[255,153,255]" pen-height="12.0" pen-width="8.0" pen-opacity="0.8"/>
<Highlighter-style name="Highlighter 2" pen-color="[255,204,0]" pen-height="14.0" pen-width="14.0" pen-opacity="0.8"/>
</Styles>
<Startup-Code startupcode=""/>
<Task-table>
	<Task-category name="&lt;default&gt;"/>
</Task-table>
<Task/>
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style="Title" layout="Title">Number of Graphs or Digraphs with <Font italic="true">n</Font> Vertices</Text-field>
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<Input><Text-field style="Author" layout="Author">David A. Harrington
University of Victoria,
Canada
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<Section collapsed="false" isCollapsible="true" drawButton="true" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title><Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field></Title>
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<Input><Text-field style="Text" layout="Normal">Maple is used to calculate a generating function for the number of graphs with different numbers of edges, for a given number of vertices.</Text-field><Text-field style="Text" layout="Normal">This follows the treatment in F. Harary, The number of linear, directed, rooted, and connected graphs, Trans. Amer. Math. Soc., 78, (1955) 445-463. <Hyperlink linktarget="https://doi.org/10.1090/S0002-9947-1955-0068198-2" tooltip="" hyperlink="true"><Font style="Hyperlink">doi: 10.1090/S0002-9947-1955-0068198-2</Font></Hyperlink></Text-field><Text-field style="Text" layout="Normal">The algorithm is b<Font encoding="UTF-8">ased on P\303\263lya's counting and the cycle index polynomial. The algorithm is described in detail for the example</Font> of <Font italic="true">n</Font> = 4 for undirected graphs (done also by Harary), which gives the generating function:</Text-field>
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<Input><Text-field style="Text" layout="Normal">indicating that (as shown) there is 1 graph with 6 edges, 1 with 5 edges, 2 with 4 edges, 3 with 3 edges, 2 with 2 edges, 1 with 1 edge and 1 with no edges. Setting <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjFGJ0YyRjxGMkY8">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjFGJ0YyRjxGMkY8</Equation> gives 11 for the total number of undirected graphs with 4 vertices. </Text-field><Text-field style="Text" layout="Normal">The basic idea is that permutations of the vertices that preserve adjacency (edges) give isomorphic graphs, which are only counted once.</Text-field><Text-field style="Text" layout="Normal">Other Maple codes for the total number, possibly more efficient, are given in the OEIS links below. Here use of the representative permutations allows a unified treatment for directed and undirected graphs. For undirected graphs the builtin GraphTheory:-NonIsomorphicGraphs command can produce these numbers or the graphs themselves.</Text-field><Text-field style="Text" layout="Normal"></Text-field>
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<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRLEdyb3VwVGhlb3J5RidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSI6RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlZGPUZA">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRLEdyb3VwVGhlb3J5RidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSI6RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlZGPUZA</Equation></Text-field>
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<Section collapsed="false" isCollapsible="true" drawButton="true" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title><Text-field style="Heading 1" layout="Heading 1">Description of Algorithm</Text-field></Title>
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<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="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">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</Equation></Text-field>
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<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKSZBc3NpZ247RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSI0RidGOUY5">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJuR0YoIiIlNyNGLg==</Equation></Text-field>
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<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">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</Equation></Text-field>
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<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUkjbW5HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JFEjMjRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=">IiND</Equation></Text-field>
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</Group>
<Group labelreference="L6692" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">Polya's cycle index polynomial can be found from the partitions of <Font italic="true">n</Font>, as is shown below. Each term corresponds to a conjugacy class of the symmetric permutation group (here <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation>), and the coefficients are the numbers of elements in that class. It is available directly using CycleIndexPolynomial in the GroupTheory package.</Text-field>
</Input>
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<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkobXN1YnN1cEdGJDYnLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjFGJy9GNlEnbm9ybWFsRictRjk2JFEiNEYnRjwvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRkMtSSNtb0dGJDYtUScmcGx1cztGJ0Y8LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZMLyUpc3RyZXRjaHlHRkwvJSpzeW1tZXRyaWNHRkwvJShsYXJnZW9wR0ZMLyUubW92YWJsZWxpbWl0c0dGTC8lJ2FjY2VudEdGTC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRmVuLUYjNigtRjk2JFEiNkYnRjwtRkc2LVExJkludmlzaWJsZVRpbWVzO0YnRjxGSkZNRk9GUUZTRlVGVy9GWlEmMC4wZW1GJy9GZ25GYW8tRiw2J0YuRjgtRjk2JFEiMkYnRjxGQUZERl1vLUklbXN1YkdGJDYlRi5GZW9GREY8RkYtRiM2KC1GOTYkUSI4RidGPEZdby1GaW82JUYuRjhGREZdby1GaW82JUYuLUY5NiRRIjNGJ0Y8RkRGPEZGLUYjNiZGZHBGXW8tRiw2J0YuRmVvRmVvRkFGREY8RkYtRiM2JkZqbkZdby1GaW82JUYuRj5GREY8Rjw=">LCwqJCkmSSJmRzYiNiMiIiIiIiVGKUYpKigiIidGKSlGJSIiI0YpJkYmNiNGLkYpRikqKCIiKUYpRiVGKSZGJjYjIiIkRilGKSomRjVGKSlGL0YuRilGKSomRixGKSZGJjYjRipGKUYp</Equation></Text-field>
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<Input><Text-field style="Text" layout="Normal">Compare the numbers in the second row of the character table. Each class has permutations with the same cycle structure, which can be seen in the representatives.
The fixed points are implicit in the permutations and cycle type, e.g,, the permutation (1,2) could be wrtten (1,2)(3)4) and has one cycle of length 2 and two fixed pints of length 1, so its cycle type could be written [1,1,2] instead of the short form [2]. Maple only accepts the short form for permutations created using Perm.</Text-field>
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<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="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">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</Equation></Text-field>
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<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">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</Equation></Text-field>
</Output>
<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">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</Equation></Text-field>
</Output>
<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">Nyc3IjckIiIjRiU3I0YlNyMiIiQ3IyIiJQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L6698" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">These cycle structures correspond to the partitions of <Font italic="true">n</Font>. These count the numbers of cycles of different lengths.
For [1,1,2], 1 appears 2 times and 2 appears 1 time, so ListTools:-Collect gives [[1,2],[2,1]].</Text-field>
</Input>
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<Input><Text-field style="Text" layout="Normal">The partitions of <Font italic="true">n</Font> index the classes, and can be used to deduce the terms in the cycle index polynomial.
For example for [1,1,2]  a representative permutation is (1)(2)(3,4) = (3,4) (Maple chose (1,2)(3)(4) = (1,2) above).</Text-field>
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<Input><Text-field style="Text" layout="Normal">The number of permutations with cycle structure <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiakYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIxRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiakYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIxRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation> cycles of length 1, <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiakYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIyRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiakYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIyRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation> of length 2 etc, is <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>.</Text-field>
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<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="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">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</Equation></Text-field>
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<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVElY29lZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKSZBc3NpZ247RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSI2RidGOUY5">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVjb2VmR0YoIiInNyNGLg==</Equation></Text-field>
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<Input><Text-field style="Text" layout="Normal">We want permutations of the vertices that send edges to edges, so we need to convert the permutations of the vertices to permutations of the edges (Harary's group <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation>).
There are 6 pairs of vertices, so 6 possible edges = binomial(4,2). The permutations in <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation> are of degree 6, and the group <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation> is isomorphic to <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation>.
Construct the possible graph edges and an index table to convert from edge to edge number.</Text-field>
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<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY2LUkjbWlHRiQ2JVEmZWRnZXNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JKG1mZW5jZWRHRiQ2Ji1GIzYoLUYsNiVRKWNvbWJpbmF0RidGL0YyLUY2Ni1RIzotRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORmVuLUYsNiVRJ2Nob29zZUYnRi9GMi1GUDYkLUYjNiYtRlA2Ji1GIzYnLUY2Ni1RIiRGJ0Y5RjtGPkZARkJGREZGRkhGWkZmbi1JI21uR0YkNiRRIjFGJ0Y5LUY2Ni1RIy4uRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnRmZuLUYsNiVRIm5GJ0YvRjJGOUY5LyUlb3BlbkdRInxmckYnLyUmY2xvc2VHUSJ8aHJGJy1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIRlovRk5RLDAuMzMzMzMzM2VtRictRmZvNiRRIjJGJ0Y5RjlGOS1GUDYmLUYjNiQtRiw2JVEhRidGL0YyRjlGOS9GYnBRIltGJy9GZXBRIl1GJ0Y5RjlGZ3FGaXEtRjY2LVEiO0YnRjlGO0ZqcEZARkJGREZGRkhGWkZNLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkhGWkZmbi1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR0Zlbi8lJmRlcHRoR0Zmci8lKmxpbmVicmVha0dRKG5ld2xpbmVGJ0Zeci1GLDYlUSRkZWdGJ0YvRjJGNS1GLDYlUSVub3BzRidGL0YyLUZQNiQtRiM2JUYrLyUrZXhlY3V0YWJsZUdGPUY5RjlGW3JGYXItRiw2JVEqZWRnZXRhYmxlRidGL0YyRjUtRiw2JVEmdGFibGVGJ0YvRjItRlA2JC1GIzYpRl5yRistRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSEZKRk0tRjY2LVEifGlyRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnL0ZORlt1LUZQNiYtRiM2J0Zib0Zlb0Zpb0Zec0Y5RjlGZ3FGaXFGaHNGOUY5LUY2Ni1RIjpGJ0Y5RjtGPkZARkJGREZGRkhGSkZNRmhzRjk=">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</Equation></Text-field>
</Input>
<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZlZGdlc0dGKDcoPCQiIiIiIiM8JEYwIiIkPCRGMCIiJTwkRjFGMzwkRjFGNTwkRjNGNTcjRi4=</Equation></Text-field>
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<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEkZGVnRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJkFzc2lnbjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjZGJ0Y5Rjk=">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRkZWdHRigiIic3I0Yu</Equation></Text-field>
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<Input><Text-field style="Text" layout="Normal">Take a permutation of the vertices; we want to convert it to a permutation on the edges
If the permutation <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JlEnJiM5NjM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGNC8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRi82JVEhRicvRjNRJXRydWVGJy9GOFEnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjVGNw==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JlEnJiM5NjM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGNC8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRi82JVEhRicvRjNRJXRydWVGJy9GOFEnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjVGNw==</Equation> takes <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> in <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation>, then its equivalent in <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation> takes the edge <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>. 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<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="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">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</Equation></Text-field>
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<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUkqbWNvbXBsZXRlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiQtSSVtcm93R0YkNiYtSSNtaUdGJDYlUSJQRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJkFzc2lnbjtGJy9GNlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTy1JKG1mZW5jZWRHRiQ2JC1GLDYoLUkjbW5HRiQ2JFEiM0YnRjwtRjk2LVEiLEYnRjxGPi9GQkY0RkNGRUZHRklGSy9GTlEmMC4wZW1GJy9GUVEsMC4zMzMzMzMzZW1GJy1GWDYkUSI0RidGPC8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJSlyZWFkb25seUdGQEY8RjxGPC1GLDYoLUYvNiVRJVBlcm1GJ0YyRjUtRjk2LVEwJkFwcGx5RnVuY3Rpb247RidGPEY+RkFGQ0ZFRkdGSUZLRmluL0ZRRmpuLUZTNiQtRiw2Ji1GUzYmLUYsNiYtRlM2JkZVRjwvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGYG9GY29GPEY8RmhwRltxRmBvRmNvRjxGPEZgb0Zjb0Y8">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJQR0YoLUklbXJvd0c2JEYmL0krbW9kdWxlbmFtZUdGKEYkNiMtSShtZmVuY2VkR0YwNiQtSSNtbkdGMDYjUSIzRigtRjg2I1EiNEYoNyNgRihJMF9tMjIyMDE0NDQ4ODIyNEdGKDYlSSN4cEc2Iy9JJ19sb2NhbEdGJkkwX24yMjIwMTQ0NDg4Mjg4R0YoSSJkRzYjL0ZFSTBfbjIyMjAxNDQ0ODgzMjBHRihJKkZJWF9TSU1QTEc2Iy9GRUkwX24yMjIwMTkzODMyNDY0R0Yo</Equation></Text-field>
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<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUkobWZlbmNlZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXJvd0dGJDYmLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJSlyZWFkb25seUdRJmZhbHNlRidGMkYyLyUlb3BlbkdRJyZsc3FiO0YnLyUmY2xvc2VHUScmcnNxYjtGJw==">NyMiIiM=</Equation></Text-field>
</Output>
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<Group labelreference="L6700" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">Apply it to the edge 5 =<Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>gives edge 4 = {2,3}, so in <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation><Font encoding="UTF-8"> 5 \342\206\222 4</Font></Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L6708" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="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">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</Equation></Text-field>
</Input>
<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUkobWZlbmNlZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXJvd0dGJDYoLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYvNiRRIjNGJ0YyLyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lKXJlYWRvbmx5R0Y7RjJGMi8lJW9wZW5HUScmbGN1YjtGJy8lJmNsb3NlR1EnJnJjdWI7Ric=">PCQiIiMiIiQ=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L6697" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">We repeat this on all the edges to see what the complete permutation is in <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation>, and find it is (2,3)(4,5)</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L6715" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY8LUkjbWlHRiQ2JVEmUGVkZ2VGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRJWVkZ2VGJ0YvRjItRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTlEsMC4zMzMzMzMzZW1GJy1GLDYlUSJQRidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5RjktRjY2LVEoJnNyYXJyO0YnRjlGO0Y+RkBGQkZERkZGSEZlbi9GTkZmbi1GLDYlUSVtYXAyRidGL0YyLUZQNiQtRiM2KS1GLDYlUSpQZXJtQXBwbHlGJ0YvRjJGV0ZpbkZXRlRGXG9GOUY5LUY2Ni1RIjpGJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHRmZuLyUmZGVwdGhHRmRwLyUqbGluZWJyZWFrR1EobmV3bGluZUYnLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkhGZW5GYW8tRiw2JVEkbWFwRidGL0YyLUZQNiQtRiM2KUYrRlctRiw2JVEmZWRnZXNGJ0YvRjJGV0ZpbkZcb0Y5RjktRjY2LVEiO0YnRjlGO0ZaRkBGQkZERkZGSEZlbkZNRl9wRlxxLUYsNiVRKVBhaXJQZXJtRidGL0YyRjUtRiw2JVElUGVybUYnRi9GMi1GUDYkLUYjNiYtRiw2JVEhRidGL0YyLUZQNiYtRiM2Ji1GLDYlUSRzZXFGJ0YvRjItRlA2JC1GIzYsLUYsNiVRKmVkZ2V0YWJsZUYnRi9GMi1GUDYmLUYjNiUtRiw2JVEiaUYnRi9GMkZcb0Y5RjkvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGV0ZbdEZccS1GNjYvUSNpbkYnLyUlYm9sZEdGMS9GM1ElYm9sZEYnLyUrZm9udHdlaWdodEdGanRGO0Y+RkBGQkZERkZGSEZlbkZhb0ZccS1GLDYlUSIlRidGL0YyRlxvRjlGOUZcb0Y5RjlGXnRGYXRGXG9GOUY5RmlxRl9wRlxxRl9wLUZgcDYmRmJwRmVwRmdwL0ZqcFElYXV0b0YnRmZyRlxvRjk=</Equation></Text-field>
</Input>
<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">Nyg8JCIiIiIiIzwkRiQiIiU8JEYkIiIkPCRGJUYnPCRGJUYpPCRGKUYn</Equation></Text-field>
</Output>
<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlQYWlyUGVybUdGKC1JJW1yb3dHNiRGJi9JK21vZHVsZW5hbWVHRihGJDYkLUkobWZlbmNlZEdGMDYkLUkjbW5HRjA2I1EiMkYoLUY4NiNRIjNGKC1GNTYkLUY4NiNRIjRGKC1GODYjUSI1Rig3I2BGKEkwX20yMjIwMTQ4Mzk3MDI0R0YoNiVJI3hwRzYjL0knX2xvY2FsR0YmSTBfbjIyMjAxNDgzOTcwODhHRihJImRHNiMvRk1JMF9uMjIyMDE0ODM5NzEyMEdGKEkqRklYX1NJTVBMRzYjL0ZNSTBfbjIyMjAxMzU3NzA4MjRHRig=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L6709" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">This may be more efficiently coded (in 1-D only) as</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L6711" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">Plist:=convert(P,'permlist',n):
PairPerm:=Perm(
  [(for i to deg do
      e:=edges[i];
      edgetable[{Plist[e[1]],Plist[e[2]]}];
    end do)
  ]);<Equation executable="true" style="2D Input" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjBlbUYnLyUmZGVwdGhHRjovJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRjY2JkY4RjtGPi9GQVElYXV0b0YnRisvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRic=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjBlbUYnLyUmZGVwdGhHRjovJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRjY2JkY4RjtGPi9GQVElYXV0b0YnRisvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRic=</Equation></Text-field>
</Input>
<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlQYWlyUGVybUdGKC1JJW1yb3dHNiRGJi9JK21vZHVsZW5hbWVHRihGJDYkLUkobWZlbmNlZEdGMDYkLUkjbW5HRjA2I1EiMkYoLUY4NiNRIjNGKC1GNTYkLUY4NiNRIjRGKC1GODYjUSI1Rig3I2BGKEkwX20yMjIwMTQ4Mzk3MDI0R0YoNiVJI3hwRzYjL0knX2xvY2FsR0YmSTBfbjIyMjAxNDgzOTcwODhHRihJImRHNiMvRk1JMF9uMjIyMDE0ODM5NzEyMEdGKEkqRklYX1NJTVBMRzYjL0ZNSTBfbjIyMjAxMzU3NzA4MjRHRig=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L6704" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">and now we find its cycle type</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L6706" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="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">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</Equation></Text-field>
</Input>
<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+STJuZXdzaG9ydGN5Y2xldHlwZUdGKDckIiIjRi83I0Yu</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L6705" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">
We want the cycle index polynomial in <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation>. We could convert all the permutations in <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation> to permutations in <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation>, create the permutation group <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation> and find its cycle index polynomial. But since the groups are isomorphic, the conjugacy classes are the same, and we need only work with one representative permutation from each class. We only need to find the cycle type for each class in <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation>, which may be different from the class in <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation>. Harary finds a formula relating the cycle types rather than working directly with the permutations.
The above calculations show the steps to find a term in the cycle index polynomial of <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation>:</Text-field><Text-field style="Text" layout="Normal">
1. Take the partition corresponding to the conjugacy class.
2. Find the number of elements in this class (coefficient of this term) in <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation>.
3. Create a representative permutation in <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation> from the cycle type given by the partition.
4. Convert it to a permutation in <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation>.</Text-field><Text-field style="Text" layout="Normal">5. Find its cycle type and use it to generate the cycle index term in <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation>.</Text-field><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal">For the above example of partition [1,1,2] we found the new cycle type was [2,2].
We need to add the missing 1's first, then find the new term is</Text-field>
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<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="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">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</Equation></Text-field>
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<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">NyYiIiJGIyIiI0Yk</Equation></Text-field>
</Output>
<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZjdGVybUdGKCwkKigiIiciIiIpJkkiZ0dGKDYjRjEiIiNGMSkmRjQ2I0Y2RjZGMUYxNyNGLg==</Equation></Text-field>
</Output>
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<Group labelreference="L6714" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">Since each edge can be present or not, the figure counting series for Polya's counting is <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>, which is substituted for <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEiaUYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y1L0Y5USdub3JtYWxGJw==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEiaUYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y1L0Y5USdub3JtYWxGJw==</Equation>.
The terms are then added and divided by <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiIUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdGTkYyRjw=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiIUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdGTkYyRjw=</Equation> to give the final generating function, with the coefficient of <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEiaUYnRjJGNUY4RjJGNUY4LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1L0Y5USdub3JtYWxGJw==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEiaUYnRjJGNUY4RjJGNUY4LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1L0Y5USdub3JtYWxGJw==</Equation> giving the number of graphs with <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn</Equation> edges.
The final result for <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjRGJ0YyRjxGMkY8">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjRGJ0YyRjxGMkY8</Equation> is <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field><Text-field style="Text" layout="Normal">
Substituting <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjFGJ0YyRjxGMkY8">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjFGJ0YyRjxGMkY8</Equation> gives the total number of graphs for <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn</Equation> vertices.</Text-field><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal">Only a small modification is required to deal with directed graphs. Instead of the edges being sets <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> they are now lists <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> and there are twice as many.</Text-field>
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<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="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">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</Equation></Text-field>
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<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEpZGlyZWRnZXNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSkmQXNzaWduO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkobWZlbmNlZEdGJDYmLUYjNjwtRlA2Ji1GIzYoLUkjbW5HRiQ2JFEiMUYnRjktRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTlEsMC4zMzMzMzMzZW1GJy1GWTYkUSIyRidGOS8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJSlyZWFkb25seUdGPUY5RjkvJSVvcGVuR1EnJmxzcWI7RicvJSZjbG9zZUdRJyZyc3FiO0YnRmZuLUZQNiYtRiM2KEZYRmZuLUZZNiRRIjNGJ0Y5RmFvRmRvRjlGOUZmb0Zpb0Zmbi1GUDYmLUYjNihGWEZmbi1GWTYkUSI0RidGOUZhb0Zkb0Y5RjlGZm9GaW9GZm4tRlA2Ji1GIzYoRl5vRmZuRlhGYW9GZG9GOUY5RmZvRmlvRmZuLUZQNiYtRiM2KEZeb0ZmbkZgcEZhb0Zkb0Y5RjlGZm9GaW9GZm4tRlA2Ji1GIzYoRl5vRmZuRmdwRmFvRmRvRjlGOUZmb0Zpb0Zmbi1GUDYmLUYjNihGYHBGZm5GWEZhb0Zkb0Y5RjlGZm9GaW9GZm4tRlA2Ji1GIzYoRmBwRmZuRl5vRmFvRmRvRjlGOUZmb0Zpb0Zmbi1GUDYmLUYjNihGYHBGZm5GZ3BGYW9GZG9GOUY5RmZvRmlvRmZuLUZQNiYtRiM2KEZncEZmbkZYRmFvRmRvRjlGOUZmb0Zpb0Zmbi1GUDYmLUYjNihGZ3BGZm5GXm9GYW9GZG9GOUY5RmZvRmlvRmZuLUZQNiYtRiM2KEZncEZmbkZgcEZhb0Zkb0Y5RjlGZm9GaW9GYW9GZG9GOUY5RmZvRmlvRjk=">LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlkaXJlZGdlc0dGKDcuNyQiIiIiIiM3JEYwIiIkNyRGMCIiJTckRjFGMDckRjFGMzckRjFGNTckRjNGMDckRjNGMTckRjNGNTckRjVGMDckRjVGMTckRjVGMzcjRi4=</Equation></Text-field>
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<Group labelreference="L6637" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input>
</Input>
</Group>
</Section>
<Section collapsed="false" isCollapsible="true" drawButton="true" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title><Text-field style="Heading 1" layout="Heading 1">Algorithm</Text-field></Title>
<Group labelreference="L6722" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">Find generating function for simple graphs with <Font italic="true">n</Font> vertices and various numbers of edges.
<Font italic="true">x</Font> is the name for the generating function variable or 1 to count all graphs with <Font italic="true">n</Font> vertices.
Optional third argument is 'undirected' (default) or 'directed'.</Text-field>
</Input>
</Group>
<Group labelreference="L6723" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal">countgraphs:=proc(n::nonnegint,x::{name,identical(1)},dirtype::{identical(undirected),identical(directed)})
  uses GroupTheory; # for the Perm operations
  local edges, edgetable,PairPerm,Plist,cyclepoly,i,j,k,P,p,partitions,
    lenpart,CT,coef,nfact,part,jj,fact,cycles,deg,e,setorlist;
  if n=0 then return 1 end if;
  nfact:=n!;
  if _npassed=3 and dirtype='directed' then
    edges := combinat:-permute(n,2);
    setorlist:=`[]`; # list operation later
  else
    edges := [combinat:-choose({$(1 .. n)},2)[]];
    setorlist:=`{}`; # set operation later
  end if;
  deg:=nops(edges);
  edgetable := table(edges=~[$(1 .. deg)]);
  cyclepoly:=0;
  # cycle through each partition of n, corresponding to a conjugacy class of SymmetricGroup(n) 
  partitions := Iterator:-Partition(n); # reverse lex order
  for part in partitions do
    # find coefficient = number in conjugate class corresponding to this partition
    # and find representative permutation in SymmetricGroup(n).
    # hand code coef and Perm calc for one pass through part
    coef:=1;jj:=0;i:=0;p:=0;
    P:=table();
    lenpart:=length(partitions); # length updates for each partition
    for k to lenpart do 
      j:=part[k];
      if j&lt;&gt;jj then jj:=j; fact:=1 else fact++ end if; # update factorial factor
      coef:=coef*j*fact;
      if j&lt;&gt;1 then
        p++;
        P[p]:=[to j do i:=i+1 od];
      end if;      
    end do:
    coef:=nfact/coef;
    P:=Perm(convert(P,'list'));
    # apply the permutation to the edges to find the permutation in the pair group
    Plist:=convert(P,'permlist',n);
    PairPerm:=Perm(
      [(for i to deg do
        e:=edges[i];
        edgetable[setorlist(Plist[e[1]],Plist[e[2]])];
      end do)
      ]);
    # find new cycletype
    CT:=PermCycleType(PairPerm);
    # 1's are suppressed in PermCycleType so we have to add them
    j:=deg-add(CT);
    cycles:=[1$j,CT[]];
    # generate the term in the cycle index polynomial for this class
    cyclepoly:=cyclepoly+coef*mul(1+x^i,i in cycles);
  end do;
  expand(cyclepoly/nfact);
end proc:</Text-field>
</Input>
</Group>
<Group labelreference="L6688" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group>
</Section>
<Section collapsed="false" isCollapsible="true" drawButton="true" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title><Text-field style="Heading 1" layout="Heading 1">Output</Text-field></Title>
<Group labelreference="L6733" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">Generating function for undirected graphs with 4 vertices, and the total numbers of undirected graphs for <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>.</Text-field>
</Input>
</Group>
<Group hide-input="false" labelreference="L6726" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="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">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</Equation></Text-field>
</Input>
<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LDAqJClJInhHNiIiIiciIiJGKCokKUYlIiImRihGKComIiIjRigpRiUiIiVGKEYoKiYiIiRGKClGJUYxRihGKComRi1GKClGJUYtRihGKEYlRihGKEYo</Equation></Text-field>
</Output>
<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKCZjb21tYTtGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnRitGMi1GLDYkUSIyRidGL0YyLUYsNiRRIjRGJ0YvRjItRiw2JFEjMTFGJ0YvRi8=">NiciIiJGIyIiIyIiJSIjNg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L6734" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal">Generating function for directed graphs with 3 vertices, and the total numbers of directed graphs for <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>.</Text-field>
</Input>
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<Section collapsed="false" isCollapsible="true" drawButton="true" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title><Text-field style="Heading 1" layout="Heading 1">Conclusions</Text-field></Title>
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<Input><Text-field style="Text" layout="Normal">Use of Maple's facility with partitions and permutations allows bypassing the machinery of the GroupTheory package (aside from the Perm manipulation routines) and leads to an efficient algorithm for the generating functions, which is only trivially different for directed and undirected graphs.</Text-field>
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<Section collapsed="false" isCollapsible="true" drawButton="true" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title><Text-field style="Heading 1" layout="Heading 1">References</Text-field></Title>
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<Input><Text-field style="Text" layout="Normal">1. F. Harary, The number of linear, directed, rooted, and connected graphs, Trans. Amer. Math. Soc., 78 (1955) 445-463. <Hyperlink linktarget="https://doi.org/10.1090/S0002-9947-1955-0068198-2" tooltip="" hyperlink="true"><Font style="Hyperlink">doi: 10.1090/S0002-9947-1955-0068198-2</Font></Hyperlink></Text-field><Text-field style="Text" layout="Normal">2. Polya Enumeration Theorem, <Hyperlink linktarget="https://en.wikipedia.org/wiki/P%C3%B3lya_enumeration_theorem" tooltip="" hyperlink="true"><Font style="Hyperlink">Wikipedia</Font></Hyperlink>.</Text-field><Text-field style="Text" layout="Normal">3. Cycle Index, <Hyperlink linktarget="https://en.wikipedia.org/wiki/Cycle_index#The_cycle_index_of_the_edge_permutation_group_of_the_complete_graph_on_four_vertices" tooltip="" hyperlink="true"><Font style="Hyperlink">Wikipedia</Font></Hyperlink> (describes the permutations of <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSI0RidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ==</Equation>)</Text-field><Text-field style="Text" layout="Normal">4. The On-Line Encyclopedia of Integer Sequences. Undirected graphs: <Hyperlink linktarget="https://oeis.org/A000088" tooltip="" hyperlink="true"><Font style="Hyperlink">OEIS A000088</Font></Hyperlink>, Directed graphs: <Hyperlink linktarget="https://oeis.org/A000273" tooltip="" hyperlink="true"><Font style="Hyperlink">OEIS A000273</Font></Hyperlink>.</Text-field><Text-field style="Text" layout="Normal"></Text-field>
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<Input><Text-field style="Text" layout="Normal"><Font encoding="UTF-8" italic="true">Legal Notice: \302\251 David A. Harrington 2023. Maplesoft and Maple are trademarks of Waterloo Maple Inc. Neither Maplesoft nor the authors are responsible for any errors contained within and are not liable for any damages resulting from the use of this material.  This application is intended for non-commercial, non-profit use only. Contact the authors for permission if you wish to use this application in for-profit activities. </Font></Text-field>
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