Permutations are among the most basic elements of discrete mathematics. They can be used to represent discrete groups of transformations and in particular play a key role in the description of the concept of symmetry. The Wolfram Language provides new functionality to work with permutations, both in list and cyclic form, and allows their action on generic expressions in a variety of ways.

Permutation Representation

Cycles — cyclic permutation representation

PermutationCyclesQ — test validity

PermutationCycles — convert to cyclic representation

PermutationList — convert to permutation list representation

PermutationListQ — test validity

RandomPermutation — random generation of permutations

Permutation Operations

PermutationReplace — standard action of a permutation on other objects

PermutationProduct  ▪  InversePermutation  ▪  PermutationPower

Permute — permute arguments of an expression

FindPermutation — permutation linking two expressions

Permutations — all permutations of arguments of an expression

Permutation Properties

PermutationOrder — order of a permutation

PermutationSupport  ▪  PermutationLength  ▪  PermutationMin  ▪  PermutationMax

Permutation Lists

Sort — return identity permutation list

Part — product of permutation lists

Ordering — inverse of a permutation list

Signature — signature of a permutation list

RandomSample — random generation of permutation lists