Permute

Move the item at position $i$ of list to position $\sigma(i)$, for a permutation perm given as Cycles or a one-line word. Given a PermutationGroup instead, returns list permuted by every element of the group.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguagePermute!↗

Permute(list, perm)list with position $i$ moved to $\sigma(i)$

CombinatoricaIf[ListQ[perm] && PermutationQ[perm] && Length[list] == Length[perm], Permute[list, InversePermutation[perm]], Missing["NotCombinatorica"]]

Permute(list, group)one permuted copy of list per element of group

CombinatoricaIf[ListQ[group] && PermutationQ[group] && Length[list] == Length[group], Permute[list, InversePermutation[group]], Missing["NotCombinatorica"]]

Typed signatures

Type	Package	Applies
(list<any>, expression<Cycles> | expression<PermutationGroup> | list<integer>) -> list<any> | list<list<any>>	groupalgebra	always

Domain: Permutations

Details

• Permute and PermutationCycles/InversePermutation all read $\sigma$ the same way: a one-line word {v1, ..., vn} means $\sigma(i) = v_i$
• A cycle or one-line word longer than list leaves the call unevaluated — there is nowhere for the extra positions to go
• The PermutationGroup form orders its output the way GroupElements orders that group

Examples

Basic #

live ↗

In$\mathrm{Permute}(\bigl\lbrack a, b, c, d\bigr\rbrack, \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 3, 2\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack b, c, a, d\bigr\rbrack$

live ↗

In$\mathrm{Permute}(\bigl\lbrack a, b, c, d, k\bigr\rbrack, \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 3, 2\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack b, c, a, d, k\bigr\rbrack$

live ↗

In$\mathrm{Permute}(\bigl\lbrack a, b, c, d, k\bigr\rbrack, \bigl\lbrack3, 1, 2\bigr\rbrack)$Out$\bigl\lbrack b, c, a, d, k\bigr\rbrack$

live ↗

In$\mathrm{Permute}(\bigl\lbrack10, 20, 30\bigr\rbrack, \bigl\lbrack2, 3, 1\bigr\rbrack)$Out$\bigl\lbrack30, 10, 20\bigr\rbrack$

live ↗

In$\mathrm{Permute}(\bigl\lbrack10, 20, 30, 40, 50\bigr\rbrack, \bigl\lbrack3, 1, 2, 5, 4\bigr\rbrack)$Out$\bigl\lbrack20, 30, 10, 50, 40\bigr\rbrack$

Scope #

live ↗

In$\mathrm{Permute}(\bigl\lbrack a, b, c, d\bigr\rbrack, \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 3\bigr\rbrack, \bigl\lbrack2, 4\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack c, d, a, b\bigr\rbrack$

live ↗

In$\mathrm{Permute}(\bigl\lbrack a, b, c, d\bigr\rbrack, \bigl\lbrack2, 3, 4, 1\bigr\rbrack)$Out$\bigl\lbrack d, a, b, c\bigr\rbrack$

Applications #

one permuted copy per element of the cyclic group of order 3

live ↗

In$\mathrm{Permute}(\bigl\lbrack a, b, c\bigr\rbrack, \mathrm{PermutationGroup}(\bigl\lbrack\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack\bigr\rbrack)\bigr\rbrack))$Out$\bigl\lbrack\bigl\lbrack a, b, c\bigr\rbrack, \bigl\lbrack c, a, b\bigr\rbrack, \bigl\lbrack b, c, a\bigr\rbrack\bigr\rbrack$

About

Examples	8
Checked against	Wolfram Language 5 agree1 differCombinatorica 2 agree
Crosswalk	Wolfram LanguageCombinatorica

Implementation

mappedcombinatoricapackages/oracle/src/mappings.ts

Combinatorica's Permute applies the inverse of the permutation ours applies. Covered for a one-line permutation of the same length, not Cycles or a padded word.

See also: Cycles, PermutationCycles, PermutationGroup, GroupElements