Perm
construct a permutation class
Calling Sequence
Parameters
Description
Examples
Compatibility
Perm( L )
Perm( LL )
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
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. In particular, the identity permutation is represented by the expression Perm([]).
The Permutation Operations in GroupTheory page lists commands that operate on permutation objects and are part of the GroupTheory package.
Note that the noncommutative multiplication operator . can be used to multiply permutations.
$a\u2254\mathrm{Perm}\left(\left[\left[1\,2\right]\,\left[3\,4\,5\right]\right]\right)$
${a}{\u2254}\left({1}{\,}{2}\right)\left({3}{\,}{4}{\,}{5}\right)$
${a}_{1}$
${2}$
${a}_{2}$
${1}$
${a}_{3}$
${4}$
${a}_{4}$
${5}$
${a}_{5}$
${3}$
$b\u2254\mathrm{Perm}\left(\left[\left[1\,3\right]\,\left[2\,6\right]\right]\right)$
${b}{\u2254}\left({1}{\,}{3}\right)\left({2}{\,}{6}\right)$
In the following examples, the PermDegree() and PermProduct() commands are part of the GroupTheory package. They operate on permutation objects constructed by Perm.
$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$
$\mathrm{PermDegree}\left(a\right)$
$\mathrm{PermDegree}\left(b\right)$
${6}$
$\mathrm{PermProduct}\left(a\,b\right)$
$\left({1}{\,}{6}{\,}{2}{\,}{3}{\,}{4}{\,}{5}\right)$
The Perm command was introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
See Also
GroupTheory
Permutation Operations in GroupTheory
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