BUILT-IN MATHEMATICA SYMBOL

PermutationCycles

PermutationCycles[perm]
gives a disjoint cycle representation of permutation perm.

The input permutation perm can be given as a permutation list or in disjoint cyclic form.

PermutationCycles[perm] returns an expression with head Cycles containing a list of cycles, each of the form , which represents the mapping of the to . The last point is mapped to .

Cyclic form of a permutation list of length 10:

Identity permutation list:

Cyclic form of a permutation list of length 10:

Out[1]=

Identity permutation list:

Out[1]=

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:

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: