({{, (RefLink[InversePermutation, paclet:ref/InversePermutation])[perm] 
returns the inverse of permutation perm.}})

OpenersNotesSection(Spacer1)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.

OpenersPrimaryExamplesSection(Spacer1)Examples  (8)

OpenersExampleSection0(Spacer1)Basic Examples  (2)

Inverse of a permutation:

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

Their product gives the identity permutation:

PermutationProduct[Cycles[{{3,2,5,1},{4,7}}],%]

Some permutations, called involutions, are their own inverse:

InversePermutation[Cycles[{{1,2},{3,4}}]]

OpenersExampleSection1(Spacer1)Scope  (1)

Invert a permutation:

InversePermutation[Cycles[{{1,10^2,10^4},{10^6,10^8}}]]

OpenersExampleSection2(Spacer1)Generalizations & Extensions  (1)

On symbolic expressions other than permutations the result is given in terms of (RefLink[PermutationPower, paclet:ref/PermutationPower]):

InversePermutation[a]

OpenersExampleSection4(Spacer1)Properties & Relations  (4)

(RefLink[InversePermutation, paclet:ref/InversePermutation]) is equivalent to (RefLink[PermutationPower, paclet:ref/PermutationPower]) with exponent -1:

InversePermutation[Cycles[{{1,6,12},{2,9}}]]==PermutationPower[Cycles[{{1,6,12},{2,9}}],-1]

Inverting a permutation is equivalent to reversing its cycles:

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:

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

PermutationOrder[perm]

InversePermutation[perm]

PermutationPower[perm,40-1]

(RefLink[Ordering, paclet:ref/Ordering]) gives the inverse of a permutation list:

perm=Cycles[{{1,9,6,5,10,7,8,4},{2,3}}];

list=PermutationList[perm]

iperm=InversePermutation[perm]

PermutationList[iperm]

Ordering[list]

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

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

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