<Chapter><Heading>Some Permutation Essentials</Heading>
In this chapter we mention a couple functions that are fairly basic but useful tools to
work with.
<Section><Heading> Complement </Heading>
<ManSection>
<Func Name="PermComplement" Arg="perm"/>
<Returns>The permutation that is the complement of <C>perm</C>.</Returns>
<Description>
The complement of a permutation <M>\tau=\tau_{1}\ldots\tau_{n}</M> is the permutation
<Display Mode="M">\tau^{C}=(n+1)-\tau_{1}\ \ (n+1)-\tau_{2}\ldots (n+1)-\tau_{n}</Display>.
<Example><![CDATA[
gap> PermComplement([3,2,8,6,7,1,5,4]);
[ 6, 7, 1, 3, 2, 8, 4, 5 ]
gap> ]]></Example>
</Description>
</ManSection>
</Section>
<Section><Heading> Rank Encoding </Heading>
<ManSection>
<Func Name="IsRankEncoding" Arg="perm"/>
<Returns><C>true</C> if <C>perm</C> is a valid rank encoding of a permutation.</Returns>
<Description>
<C>IsRankEncoding</C> checkes whether the input list <C>perm</C> is a valid rank encoding
by checking whether it is accepted by the bounded class automaton, with the highest rank
being set by the highest element in <C>perm</C>.
<Example><![CDATA[
gap> IsRankEncoding([3,2,6,4,4,1,2,1]);
true
gap> IsRankEncoding([3,2,6,4,5,1,2,1]);
false
gap> ]]></Example>
</Description>
</ManSection>
</Section>
<!--<Section><Heading> Acceptance </Heading> --> <!--<ManSection> --> <!-- <Func Name="IsAcceptedWord" Arg="aut,perm"/> --> <!-- <Returns><C>true</C> if <C>perm</C> is accepted by <C>aut</C>.</Returns> --> <!-- <Description> --> <!-- <C>IsAcceptedWord</C> checkes whether the input rank encoded <C>perm</C> is an accepted --> <!-- permutation of <C>aut</C>. What makes <C>IsAcceptedWord</C> special is that it --> <!-- translates the input list into a rational expression in terms of the appropriate alphabet --> <!-- for the automata used in <C>PatternClass</C>. --> <!--<Example><![CDATA[ --> <!--gap> a:=MinimalAutomaton(GraphToAut(Seqstacks(2,2),1,6)); --> <!--< deterministic automaton on 3 letters with 3 states > --> <!--gap> IsAcceptedWord(a,[ 3, 3, 3, 1, 3, 1, 2, 1 ]); --> <!--true --> <!--gap> ]]></Example> --> <!-- </Description> --> <!--</ManSection> --> <!--</Section> -->
</Chapter>
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