PermutationCycles

PermutationCycles[perm]

gives a disjoint cycle representation of permutation perm.

Details

  • The input permutation perm can be given as a permutation list or in disjoint cyclic form.
  • A permutation list is a reordering of the consecutive integers {1,2,…,n}.
  • PermutationCycles[perm] returns an expression with head Cycles containing a list of cycles, each of the form {p1,p2,…,pn}, which represents the mapping of the pi to pi+1. The last point pn is mapped to p1.
  • PermutationCycles[perm,h] returns an expression with head h.
  • The result of PermutationCycles is automatically canonicalized by rotating each cycle so that the smallest point appears first and ordering cycles by the first point.

Examples

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Basic Examples  (2)

Cyclic form of a permutation list of length 10:

Identity permutation list:

Scope  (4)

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:

Applications  (2)

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

Find the signature of a permutation list:

Properties & Relations  (6)

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 permutation matrix corresponding to a given permutation list:

Use the "PermutationCycles" property of PermutationMatrix to get the corresponding disjoint cycle representation:

This is equivalent to directly applying PermutationCycles to the permutation list:

A Wolfram Language 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:

This is equivalent to FactorialPower:

Compute the factorization:

Its coefficients are Stirling numbers of the first kind:

Neat Examples  (1)

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

Wolfram Research (2010), PermutationCycles, Wolfram Language function, https://reference.wolfram.com/language/ref/PermutationCycles.html (updated 2012).

Text

Wolfram Research (2010), PermutationCycles, Wolfram Language function, https://reference.wolfram.com/language/ref/PermutationCycles.html (updated 2012).

CMS

Wolfram Language. 2010. "PermutationCycles." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2012. https://reference.wolfram.com/language/ref/PermutationCycles.html.

APA

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

BibTeX

@misc{reference.wolfram_2024_permutationcycles, author="Wolfram Research", title="{PermutationCycles}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/PermutationCycles.html}", note=[Accessed: 18-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_permutationcycles, organization={Wolfram Research}, title={PermutationCycles}, year={2012}, url={https://reference.wolfram.com/language/ref/PermutationCycles.html}, note=[Accessed: 18-November-2024 ]}