MajorIndexMinusInversions

The major index less the number of inversions.

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MajorIndexMinusInversions(p)The major index less the number of inversions.

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(Filter(
                                     Range(1, Length(_x) - 1),
                                     Function(_x[i] > _x[i + 1], i),
                                   )) - Sum(Map(
                                                Function(Count(Filter(
                                                                       Range(i +
                                                                             1, Length(_x)),
                                                                       Function(
                                                                                 _x[i] > _x[j],
                                                                                 j,
                                                                               ),
                                                                     )), i),
                                                Range(1, Length(_x)),
                                              ))

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001377

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001377

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001377

About

Examples	6
Crosswalk	FindStat