Perm  construct a permutation class

Calling Sequence


Perm( L )
Perm( LL )


Parameters


L



list(posint) : a list of positive integers that forms a permutation of 1..n, for some n

LL



list(list(posint)) : a list of lists of positive integers representing disjoint cycles





Description


•

A permutation is a bijective mapping from the set$\left\{1\,2\,\dots \,n\right\}$ to itself, for some positive integer $n$.

•

The set of all such permutations forms the symmetric group of degree $n$, and subgroups of symmetric groups are permutation groups.

•

Permutations are typically represented as products of disjoint cycles, each of which is an orbit of the permutation. This is a list of the form$\left[{c}_{1}\,{c}_{2}\,\dots \,{c}_{k}\right]$ in which each ${c}_{i}$ is itself a list$\left[{i}_{1}\,{i}_{2}\,\dots \,{i}_{m}\right]$ representing a cycle of the form${i}_{1}\mapsto {i}_{2}\mapsto {i}_{m}\mapsto {i}_{1}$ .

•

The Perm constructor creates a permutation, given a specification of its disjoint cycle structure in the form of a list of lists. You can also use a permutation list, which is just the representation of the permutation as a list L of points in which L[ i ] specifies the image of i under the permutation.

•

Note that the noncommutative multiplication operator . can be used to multiply permutations.



Compatibility


•

The Perm command was introduced in Maple 17.



Examples


>

a := Perm( [ [ 1, 2 ], [ 3, 4, 5 ] ] );

${a}{:=}\left({1}{\,}{2}\right)\left({3}{\,}{4}{\,}{5}\right)$
 (1) 
>

a[ 1 ]; # the image of 1 under a

>

a[ 2 ]; # the image of 2 under a

>

a[ 3 ]; # the image of 3 under a

>

a[ 4 ]; # the image of 4 under a

>

a[ 5 ]; # the image of 5 under a

>

b := Perm( [ [ 1, 3 ], [ 2, 6 ] ] );

${b}{:=}\left({1}{\,}{3}\right)\left({2}{\,}{6}\right)$
 (7) 
In the following examples, the PermDegree() and PermProduct() commands are part of the GroupTheory package. They operate on permutation objects constructed by Perm.
$\left({1}{\,}{6}{\,}{2}{\,}{3}{\,}{4}{\,}{5}\right)$
 (10) 


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