Cyclic form of a permutation list of length 10:

Identity permutation list:

Action on permutation lists:

With a head other than Cycles , singletons are kept:

On other cyclic permutations the input is returned unchanged:

PermutationCycles works efficiently with large permutation lists:

Permutation cycles can be considered a sparse representation of permutation lists:

Find the signature of a permutation list:

The permutation returned by PermutationCycles [list] produces with Permute the same result as using the original list with Part :

The collection of cycles returned by PermutationCycles corresponds to the permutation that generates the list from sorted order:

PermutationList gives the inverse of PermutationCycles :

A combination of PermutationCycles and PermutationList adds singletons:

A Mathematica implementation of PermutationCycles :

The built-in version is faster:

Number of permutations of the symmetric group with 6 to 1 cycles, including 1-cycles:

Construct an associated polynomial:

Compute the factorization:

Its coefficients are Stirling numbers of the first kind:

Average number of cycles for permutation lists of increasing length. Compare with the theoretical estimate: