CycleDecomposition

The singular-inhabitant constructor for a permutation in cycle notation with its fixed points kept: each cycle from its least point, cycles in order of those points.

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CycleDecomposition(cycles)The singular-inhabitant constructor for a permutation in cycle notation with its fixed points kept: each cycle from its least point, cycles in order of those points.

CycleDecomposition(cycles, n)the permutation of 1..n that a Cycles value describes, the points it leaves out kept as fixed points

Typed signatures

Type	Package	Applies
((list<list<integer>>) -> cycle_decomposition) & ((permutation) -> cycle_decomposition) & ((expression<Cycles>, integer) -> cycle_decomposition)	combinatorics	always

Domain: Collections

Details

• Keeping the fixed points is what makes it a permutation of a definite size, so it converts back: Permutation(CycleDecomposition(p)) is p, and every permutation statistic answers on a decomposition. Wolfram's Cycles drops them, so it can't say how many points there are.
• CycleDecomposition(Permutation([…])) converts; CycleDecomposition of anything else holds it as given.
• CycleDecomposition(Cycles(...), n) puts the fixed points back, and each cycle from its least point: Cycles doesn't know its size, so n says it, as Wolfram's PermutationList(c, n) does. It declines a point outside $1..n$ or one that appears twice.
• Cycles(CycleDecomposition(...)) is the other direction, and forgets the fixed points. The two round-trip: CycleDecomposition(Cycles(c), n) is c for any decomposition c of size $n$.

Maps

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

From CycleDecomposition

• PermutationCycleDecomposition → PermutationinverseThe permutation a cycle decomposition describes.

To CycleDecomposition

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

Examples

Basic #

From a permutation: $4$ is a fixed point, kept as its own cycle

live ↗

In$\operatorname{CycleDecomposition}({\operatorname{Permutation}}([2, 3, 1, 4]))$Out${\operatorname{CycleDecomposition}}([[1, 2, 3], [4]])$

And back

live ↗

In$\operatorname{Permutation}({\operatorname{CycleDecomposition}}([[1, 3], [2]]))$Out${\operatorname{Permutation}}([3, 2, 1])$

A permutation statistic, through the conversion

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{CycleDecomposition}}([[1, 2, 3], [4]]), \text{FixedPoints})$Out$1$

From Cycles, which drops the fixed points: the size $4$ puts them back

live ↗

In$\operatorname{CycleDecomposition}(\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack2, 3\bigr\rbrack\bigr\rbrack), 4)$Out${\operatorname{CycleDecomposition}}([[1], [2, 3], [4]])$

A cycle may be written from any of its points: each comes out from its least

live ↗

In$\operatorname{CycleDecomposition}(\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack3, 1, 2\bigr\rbrack\bigr\rbrack), 4)$Out${\operatorname{CycleDecomposition}}([[1, 2, 3], [4]])$

To Cycles, the fixed points forgotten

live ↗

In$\mathrm{Cycles}({\operatorname{CycleDecomposition}}([[1], [2, 3], [4]]))$Out$\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack2, 3\bigr\rbrack\bigr\rbrack)$

About

Examples	6
Checked against	Wolfram Language 3 agree2 differSymPy 2 differmpmath 2 differ
Crosswalk	Wikipedia ×2MathWorldWikidataEncyclopedia of MathematicsnLabBritannica

See also: PermutationsAsCycles, Cycles, PermutationCycles