PermutationsAsCycles

Every permutation of $\{1, …, n\}$ in cycle notation, fixed points kept, listed by cycle type.

PermutationsAsCycles(n)every permutation of

\{1, …, n\}

as a cycle decomposition, by cycle type

Typed signatures

Type	Package	Applies
`(integer<0..>) -> indexed_collection<cycle_decomposition>`	`combinatorics`	always

Domain: Collections

Details

The same permutations as SymmetricGroup, in another order: by cycle type, in IntegerPartitions' order (the $n$ -cycles first, the identity last), then by the canonical cycle form, lexicographically. The order is our choice, not something cycle notation implies.
Each element is a CycleDecomposition, so every permutation statistic answers on it.

Maps

Into and out of CycleDecomposition, called as CombinatorialMap(x, "Name"). Properties are the laws we state and what FindStat records.

PermutationCycleDecomposition → PermutationinverseThe permutation a cycle decomposition describes.

CycleDecompositionPermutation → CycleDecompositioninverseA permutation in cycle notation, fixed points kept: each cycle from its least point, cycles in order of those points.

1 more kept as data, checked against the oracles

Basic #

$4!$

\mathrm{Count}(\mathrm{PermutationsAsCycles}(4))

Out

24

The first is an $n$ -cycle; the identity comes last

\mathrm{PermutationsAsCycles}(4)[1]

Out

{\operatorname{CycleDecomposition}}([[1, 2, 3, 4]])

Statistics #

A permutation statistic, through the conversion: the Stirling numbers of the first kind

\mathrm{Tally}(\mathrm{CombinatorialStat}(\mathrm{PermutationsAsCycles}(4), \text{CycleCount}))

Out

\operatorname{Tuple}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack,\bigl\lbrack6, 11, 6, 1\bigr\rbrack)

About

Examples	4
Crosswalk	OEIS