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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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<Chapter><Heading>Some Permutation Essentials</Heading>

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In this chapter we mention a couple functions that are fairly basic but useful tools to 

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work with.

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<Section><Heading> Complement </Heading> 

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<ManSection>

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  <Func Name="PermComplement" Arg="perm"/>

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  <Returns>The permutation that is the complement of <C>perm</C>.</Returns>

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  <Description>

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    The complement of a permutation <M>\tau=\tau_{1}\ldots\tau_{n}</M> is the permutation 

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    <Display Mode="M">\tau^{C}=(n+1)-\tau_{1}\ \ (n+1)-\tau_{2}\ldots (n+1)-\tau_{n}</Display>.

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<Example><![CDATA[

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gap> PermComplement([3,2,8,6,7,1,5,4]);

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[ 6, 7, 1, 3, 2, 8, 4, 5 ]

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gap> ]]></Example>

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  </Description>

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</ManSection>

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</Section>

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<Section><Heading> Rank Encoding </Heading> 

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<ManSection>

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  <Func Name="IsRankEncoding" Arg="perm"/>

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  <Returns><C>true</C> if <C>perm</C> is a valid rank encoding of a permutation.</Returns>

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  <Description>

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    <C>IsRankEncoding</C> checkes whether the input list <C>perm</C> is a valid rank encoding

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    by checking whether it is accepted by the bounded class automaton, with the highest rank

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    being set by the highest element in <C>perm</C>.

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<Example><![CDATA[

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gap> IsRankEncoding([3,2,6,4,4,1,2,1]);

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true

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gap> IsRankEncoding([3,2,6,4,5,1,2,1]);

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false

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gap> ]]></Example>

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  </Description>

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</ManSection>

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</Section>

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<!--<Section><Heading> Acceptance </Heading> -->

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<!--<ManSection> -->

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<!--  <Func Name="IsAcceptedWord" Arg="aut,perm"/> -->

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<!--  <Returns><C>true</C> if <C>perm</C> is accepted by <C>aut</C>.</Returns> -->

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<!--  <Description> -->

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<!--    <C>IsAcceptedWord</C> checkes whether the input rank encoded <C>perm</C> is an accepted -->

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<!--    permutation of <C>aut</C>. What makes <C>IsAcceptedWord</C> special is that it -->

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<!--    translates the input list into a rational expression in terms of the appropriate alphabet -->

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<!--    for the automata used in <C>PatternClass</C>. -->

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<!--<Example><![CDATA[ -->

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<!--gap> a:=MinimalAutomaton(GraphToAut(Seqstacks(2,2),1,6));        -->

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<!--< deterministic automaton on 3 letters with 3 states > -->

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<!--gap> IsAcceptedWord(a,[ 3, 3, 3, 1, 3, 1, 2, 1 ]);               -->

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<!--true -->

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<!--gap> ]]></Example> -->

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<!--  </Description> -->

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<!--</ManSection> -->

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<!--</Section> -->

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</Chapter>

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