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PermutationPower[perm,n]

gives the n^(th) permutation power of the permutation perm.

Details
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Examples  
Basic Examples  
Scope  
Generalizations & Extensions  
Properties & Relations  
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PermutationPower

PermutationPower[perm,n]

gives the n^(th) permutation power of the permutation perm.

Details

  • PermutationPower[perm,n] effectively computes the product of a permutation perm with itself n times.
  • When n is negative, PermutationPower finds powers of the inverse of the permutation perm.
  • PermutationPower[perm,0] gives the identity permutation.

Examples

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Basic Examples  (3)

Sixth power of a permutation:

Wolfram Language code: PermutationPower[Cycles[{{4, 2, 5}, {6, 3, 1, 7}}], 6]

Second power of the inverse permutation:

Wolfram Language code: PermutationPower[Cycles[{{4, 2, 5}, {6, 3, 1, 7}}], -2]

PermutationPower can yield the identity permutation:

Wolfram Language code: PermutationPower[Cycles[{{4, 2, 5}, {6, 3, 1, 7}}], 12]

Scope  (1)

Compute arbitrary powers of a permutation:

Wolfram Language code: perm = Cycles[{{1, 11, 4}, {2, 18, 10, 14}, {3, 17, 20, 13, 12, 5, 15, 19, 9, 16}, {6, 8, 7}}];
Wolfram Language code: PermutationPower[perm, 12 ^ 345]

Generalizations & Extensions  (2)

PermutationPower does not evaluate for symbolic arguments:

Wolfram Language code: PermutationPower[h, 2]
Wolfram Language code: PermutationPower[Cycles[{{3, 4}, {7, 10, 20}}], x]

PermutationPower performs some simplifications for generic symbolic input:

Wolfram Language code: PermutationPower[PermutationPower[x, 2], 5]
Wolfram Language code: PermutationPower[PermutationProduct[a, b], 3]

Properties & Relations  (1)

For exponents that are multiples of the order of the permutation, the permutation power yields identity:

Wolfram Language code: perm = Cycles[{{1, 11, 4}, {2, 18, 10, 14}, {3, 17, 20, 13, 12, 5, 15}, {6, 8, 7}}];
Wolfram Language code: PermutationPower[perm, PermutationOrder[perm]]

Hence large powers can be reduced by using the modulo of the exponent:

Wolfram Language code: PermutationPower[perm, 10 ^ 10] == PermutationPower[perm, Mod[10 ^ 10, PermutationOrder[perm]]]
Wolfram Research (2010), PermutationPower, Wolfram Language function, https://reference.wolfram.com/language/ref/PermutationPower.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_permutationpower, author="Wolfram Research", title="{PermutationPower}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/PermutationPower.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_permutationpower, organization={Wolfram Research}, title={PermutationPower}, year={2010}, url={https://reference.wolfram.com/language/ref/PermutationPower.html}, note=[Accessed: 13-June-2026]}