MatrixInversionNumber

The sum of a(i, j) a(k, l) over the pairs of entries with i > k and j < l.

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MatrixInversionNumber(tableau)The sum of a(i, j) a(k, l) over the pairs of entries with i > k and j < l.

Typed signatures

Type	Package	Applies
(alternating_sign_matrix) -> number	combinatorics	always

Domain: Alternating sign matrix statistics

Details

• Defined over AlternatingSignMatrix as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a AlternatingSignMatrix 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.
• FindStat's St000067. On a permutation matrix it is the number of inversions.

Definition

Epsil, as the implementation runs it.

on AlternatingSignMatrix, over _x

0 if Length(_x) < 1 else Apply((cells: list<integer>) => Apply((side: integer) => Max(Sum(Map(Function(cells[u] *
                                                                                                       Max(Sum(Map(
                                                                                                                    Function(cells[v] if (v -
                                                                                                                                          ((v - 1) %
                                                                                                                                           side +
                                                                                                                                           1)) /
                                                                                                                                         side +
                                                                                                                                         1 <
                                                                                                                                         (u -
                                                                                                                                          ((u - 1) %
                                                                                                                                           side +
                                                                                                                                           1)) /
                                                                                                                                         side +
                                                                                                                                         1 &&
                                                                                                                                         (v - 1) %
                                                                                                                                         side +
                                                                                                                                         1 >
                                                                                                                                         (u - 1) %
                                                                                                                                         side +
                                                                                                                                         1 else 0, v),
                                                                                                                    Range(1, side * side),
                                                                                                                  )), 0), u), Range(1, side * side))), 0), Length(_x)), Flatten(_x))

Examples

Basic #

the alternating sign matrix $[[0,1,0],[1,-1,1],[0,1,0]]$

live ↗

In$\mathrm{MatrixInversionNumber}({\operatorname{AlternatingSignMatrix}}([[0, 1, 0], [1, -1, 1], [0, 1, 0]]))$Out$2$

Properties #

FindStat's value for [[1]]

live ↗

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

Known value · FindStat St000067

FindStat's value for [[0,1],[1,0]]

live ↗

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

Known value · FindStat St000067

FindStat's value for [[1,0],[0,1]]

live ↗

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

Known value · FindStat St000067

About

Examples	4
Crosswalk	FindStat