InversionsOfDistanceAtMostTwo

Inversions i < j with j - i <= 2.

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InversionsOfDistanceAtMostTwo(p)Inversions i < j with j - i <= 2.

Typed signatures

Type	Package	Applies
(list<integer> | 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 also a bare list of integers: this reading compares entries with each other rather than with their positions, so it stands on any sequence.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

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

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $312$, as a plain list

live ↗

In$\mathrm{InversionsOfDistanceAtMostTwo}(\bigl\lbrack3, 1, 2\bigr\rbrack)$Out$2$

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000495

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000495

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

live ↗

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

Known value · FindStat St000495

About

Examples	6
Crosswalk	FindStat