WOLFRAM

Wolfram Language & System Documentation Center

InversePermutation[perm]

returns the inverse of permutation perm.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Generalizations & Extensions  
Properties & Relations  
See Also
Tech Notes
Related Guides
History
Cite this Page

InversePermutation

InversePermutation[perm]

returns the inverse of permutation perm.

Details

  • The product of a permutation with its inverse gives the identity permutation.
  • Every permutation has a uniquely defined inverse.
  • The support of a permutation is the same as the support of its inverse.

Examples

open all close all

Basic Examples  (2)

Inverse of a permutation:

Wolfram Language code: InversePermutation[Cycles[{{3, 2, 5, 1}, {4, 7}}]]

Their product gives the identity permutation:

Wolfram Language code: PermutationProduct[Cycles[{{3, 2, 5, 1}, {4, 7}}], %]

Some permutations, called involutions, are their own inverse:

Wolfram Language code: InversePermutation[Cycles[{{1, 2}, {3, 4}}]]

Scope  (1)

Invert a permutation:

Wolfram Language code: InversePermutation[Cycles[{{1, 10 ^ 2, 10 ^ 4}, {10 ^ 6, 10 ^ 8}}]]

Generalizations & Extensions  (1)

On symbolic expressions other than permutations the result is given in terms of PermutationPower:

Wolfram Language code: InversePermutation[a]

Properties & Relations  (4)

InversePermutation is equivalent to PermutationPower with exponent -1:

Wolfram Language code: InversePermutation[Cycles[{{1, 6, 12}, {2, 9}}]] == PermutationPower[Cycles[{{1, 6, 12}, {2, 9}}], -1]

Inverting a permutation is equivalent to reversing its cycles:

Wolfram Language code: InversePermutation[Cycles[{{1, 6, 12}, {2, 9}}]] == Map[Reverse, Cycles[{{1, 6, 12}, {2, 9}}], {2}]

For a permutation of finite degree, its inverse can always be obtained as the power with a positive integer:

Wolfram Language code: perm = Cycles[{{1, 13, 3, 9, 5, 4, 14, 2}, {6, 12}, {7, 11, 8, 10, 15}}];
Wolfram Language code: PermutationOrder[perm]
Wolfram Language code: InversePermutation[perm]
Wolfram Language code: PermutationPower[perm, 40 - 1]

Ordering gives the inverse of a permutation list:

Wolfram Language code: perm = Cycles[{{1, 9, 6, 5, 10, 7, 8, 4}, {2, 3}}];
Wolfram Language code: list = PermutationList[perm]
Wolfram Language code: iperm = InversePermutation[perm]
Wolfram Language code: PermutationList[iperm]
Wolfram Language code: Ordering[list]
Wolfram Research (2010), InversePermutation, Wolfram Language function, https://reference.wolfram.com/language/ref/InversePermutation.html.

Text

Wolfram Research (2010), InversePermutation, Wolfram Language function, https://reference.wolfram.com/language/ref/InversePermutation.html.

CMS

Wolfram Language. 2010. "InversePermutation." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/InversePermutation.html.

APA

Wolfram Language. (2010). InversePermutation. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InversePermutation.html

BibTeX

@misc{reference.wolfram_2026_inversepermutation, author="Wolfram Research", title="{InversePermutation}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/InversePermutation.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_inversepermutation, organization={Wolfram Research}, title={InversePermutation}, year={2010}, url={https://reference.wolfram.com/language/ref/InversePermutation.html}, note=[Accessed: 12-June-2026]}