Recoils

Values i with i + 1 to the left of i: the descents of the inverse.

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Recoils(p)Values i with i + 1 to the left of i: the descents of the inverse.

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.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

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

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000354

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000354

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

live ↗

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

Known value · FindStat St000354

About

Examples	5
Crosswalk	FindStat