({{, (RefLink[PermutationPower, paclet:ref/PermutationPower])[perm,n] 
gives the n(^th) permutation power of the permutation perm.}})

OpenersNotesSection(Spacer1)Details

(RefLink[PermutationPower, paclet:ref/PermutationPower])[perm,n] effectively computes the product of a permutation perm with itself n times.

When n is negative, (RefLink[PermutationPower, paclet:ref/PermutationPower]) finds powers of the inverse of the permutation perm.

(RefLink[PermutationPower, paclet:ref/PermutationPower])[perm,0] gives the identity permutation.

OpenersPrimaryExamplesSection(Spacer1)Examples  (7)

OpenersExampleSection0(Spacer1)Basic Examples  (3)

Sixth power of a permutation:

PermutationPower[Cycles[{{4,2,5},{6,3,1,7}}],6]

Second power of the inverse permutation:

PermutationPower[Cycles[{{4,2,5},{6,3,1,7}}],-2]

(RefLink[PermutationPower, paclet:ref/PermutationPower]) can yield the identity permutation:

PermutationPower[Cycles[{{4,2,5},{6,3,1,7}}],12]

OpenersExampleSection1(Spacer1)Scope  (1)

Compute arbitrary powers of a permutation:

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

PermutationPower[perm,12^345]

OpenersExampleSection2(Spacer1)Generalizations & Extensions  (2)

(RefLink[PermutationPower, paclet:ref/PermutationPower]) does not evaluate for symbolic arguments:

PermutationPower[h,2]

PermutationPower[Cycles[{{3,4},{7,10,20}}],x]

(RefLink[PermutationPower, paclet:ref/PermutationPower]) performs some simplifications for generic symbolic input:

PermutationPower[PermutationPower[x,2],5]

PermutationPower[PermutationProduct[a,b],3]

OpenersExampleSection4(Spacer1)Properties & Relations  (1)

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

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

PermutationPower[perm,PermutationOrder[perm]]

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

PermutationPower[perm,10^10] == PermutationPower[perm, Mod[10^10, PermutationOrder[perm]]]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[Cycles, paclet:ref/Cycles])  ▪  (RefLink[PermutationProduct, paclet:ref/PermutationProduct])  ▪  (RefLink[InversePermutation, paclet:ref/InversePermutation])}})}})

({{DocumentationFooterIconsRelatedTutorial.png, ({{Tutorials}, {▪(RefLinkPlain[Permutations, paclet:tutorial/Permutations])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Permutations, paclet:guide/Permutations])}})}})