CycleAsOneLinePreimageSize

The permutations whose cycles, each from its least point and in order of those, read p as one word.

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CycleAsOneLinePreimageSize(p)The permutations whose cycles, each from its least point and in order of those, read p as one word.

Typed signatures

Type	Package	Applies
(permutation) -> number	combinatorics	always

Domain: Permutation statistics

Details

• Defined over Permutation as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a Permutation and nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer.
• A cycle may start at each right-to-left minimum after the first entry, which must be 1.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

1 if Length(_x) < 1 else 2 ^ (Count(Filter(
                                            Range(1, Length(_x)),
                                            Function(_x[i] == Min(Map(
                                                                       Function(_x[j], j),
                                                                       Range(i, Length(_x)),
                                                                     )), i),
                                          )) - 1) if _x[1] == 1 else 0

Examples

Basic #

the one-line word $312$

live ↗

In$\mathrm{CycleAsOneLinePreimageSize}({\operatorname{Permutation}}([3, 1, 2]))$Out$0$

the one-line word $2413$

live ↗

In$\mathrm{CycleAsOneLinePreimageSize}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$0$

Properties #

FindStat's value for [1,2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2]), \text{CycleAsOneLinePreimageSize})$Out$2$

Known value · FindStat St000279

FindStat's value for [2,1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{CycleAsOneLinePreimageSize})$Out$0$

Known value · FindStat St000279

FindStat's value for [1,2,3]

live ↗

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

Known value · FindStat St000279

About

Examples	5
Crosswalk	FindStat