Perm - Maple Programming Help

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${i}_{1}↦{i}_{2}↦{i}_{m}↦{i}_{1}$ .
 • 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 non-commutative multiplication operator . can be used to multiply permutations.

Examples

 > $a≔\mathrm{Perm}\left(\left[\left[1,2\right],\left[3,4,5\right]\right]\right)$
 ${a}{≔}\left({1}{,}{2}\right)\left({3}{,}{4}{,}{5}\right)$ (1)
 > ${a}_{1}$
 ${2}$ (2)
 > ${a}_{2}$
 ${1}$ (3)
 > ${a}_{3}$
 ${4}$ (4)
 > ${a}_{4}$
 ${5}$ (5)
 > ${a}_{5}$
 ${3}$ (6)
 > $b≔\mathrm{Perm}\left(\left[\left[1,3\right],\left[2,6\right]\right]\right)$
 ${b}{≔}\left({1}{,}{3}\right)\left({2}{,}{6}\right)$ (7)

In the following examples, the PermDegree() and PermProduct() commands are part of the GroupTheory package. They operate on permutation objects constructed by Perm.

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{PermDegree}\left(a\right)$
 ${5}$ (8)
 > $\mathrm{PermDegree}\left(b\right)$
 ${6}$ (9)
 > $\mathrm{PermProduct}\left(a,b\right)$
 $\left({1}{,}{6}{,}{2}{,}{3}{,}{4}{,}{5}\right)$ (10)

Compatibility

 • The Perm command was introduced in Maple 17.