PermutationCycles

A permutation given as a one-line word $\{\sigma(1), \dots, \sigma(n)\}$, converted to disjoint-cycle notation (Cycles). The other direction of Permute's two conventions.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguagePermutationCycles✓↗

PermutationCycles(perm)the cycles of the one-line permutation perm, fixed points dropped

SageMathenumeratio_permutation_cycles(perm)

PermutationCycles(perm, f)every position, fixed points included, each wrapped as a singleton — with f used in place of Cycles

Typed signatures

Type	Package	Applies
(expression<Cycles> | list<integer>, any?) -> any	groupalgebra	always

Domain: Permutations

Details

• Cycles come out ordered by, and each starting at, its smallest point
• Already-Cycles input passes straight through unchanged — PermutationCycles is idempotent on its own output
• The identity permutation has no non-trivial cycles, so PermutationCycles({1, ..., n}) is Cycles({})

Examples

Basic #

positions 3, 7 and 10 are fixed and drop out

live ↗

In$\mathrm{PermutationCycles}(\bigl\lbrack2, 5, 3, 6, 1, 8, 7, 9, 4, 10\bigr\rbrack)$Out$\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2, 5\bigr\rbrack, \bigl\lbrack4, 6, 8, 9\bigr\rbrack\bigr\rbrack)$

the identity fixes everything

live ↗

In$\mathrm{PermutationCycles}(\bigl\lbrack1, 2, 3, 4, 5\bigr\rbrack)$Out$\mathrm{Cycles}(\bigl\lbrack\bigr\rbrack)$

Scope #

with a second argument, fixed points stay as singletons and head replaces Cycles

live ↗

In$\mathrm{PermutationCycles}(\bigl\lbrack2, 5, 3, 6, 1, 8, 7, 9, 4, 10\bigr\rbrack, \mathrm{head})$Out$\mathrm{head}(\bigl\lbrack\bigl\lbrack1, 2, 5\bigr\rbrack, \bigl\lbrack3\bigr\rbrack, \bigl\lbrack4, 6, 8, 9\bigr\rbrack, \bigl\lbrack7\bigr\rbrack, \bigl\lbrack10\bigr\rbrack\bigr\rbrack)$

already in cycle notation — passed through

live ↗

In$\mathrm{PermutationCycles}(\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 3, 5\bigr\rbrack, \bigl\lbrack2, 4, 6\bigr\rbrack\bigr\rbrack))$Out$\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 3, 5\bigr\rbrack, \bigl\lbrack2, 4, 6\bigr\rbrack\bigr\rbrack)$

Converts a one-line permutation to cycles, dropping fixed points

live ↗

In$\mathrm{PermutationCycles}(\bigl\lbrack6, 3, 2, 5, 4, 1\bigr\rbrack)$Out$\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 6\bigr\rbrack, \bigl\lbrack2, 3\bigr\rbrack, \bigl\lbrack4, 5\bigr\rbrack\bigr\rbrack)$

live ↗

In$\mathrm{PermutationCycles}(\bigl\lbrack1, 2, 3, 4, 5\bigr\rbrack, \mathrm{Identity})$Out$\bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack2\bigr\rbrack, \bigl\lbrack3\bigr\rbrack, \bigl\lbrack4\bigr\rbrack, \bigl\lbrack5\bigr\rbrack\bigr\rbrack$

Applications #

Only the two nontrivial cycles survive among twenty points

live ↗

In$\mathrm{PermutationCycles}(\bigl\lbrack2, 3, 1, 4, 5, 6, 7, 8, 9, 10, 12, 11, 13, 14, 15, 16, 17, 18, 19, 20\bigr\rbrack)$Out$\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack11, 12\bigr\rbrack\bigr\rbrack)$

Properties #

Cycles are written starting from their smallest point

live ↗

In$\mathrm{PermutationCycles}(\bigl\lbrack4, 5, 1, 3, 2\bigr\rbrack)$Out$\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 4, 3\bigr\rbrack, \bigl\lbrack2, 5\bigr\rbrack\bigr\rbrack)$

About

Examples	8
Checked against	Wolfram Language 8 agreeSageMath 6 agree
Crosswalk	Wolfram LanguageSageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts

See also: Cycles, CycleDecomposition, Permute, InversePermutation