TopLeftZeroSquareSize

The side of the largest square of zeros at the top left corner.

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TopLeftZeroSquareSize(tableau)The side of the largest square of zeros at the top left corner.

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 St001981. The square of side z is all zeros exactly when z is below the largest of the row and column of every nonzero entry, so the side is the least such largest, less 1; an alternating sign matrix has a nonzero entry.

Definition

Epsil, as the implementation runs it.

on AlternatingSignMatrix, over _x

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

Examples

Basic #

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

live ↗

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

Properties #

FindStat's value for [[1]]

live ↗

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

Known value · FindStat St001981

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

live ↗

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

Known value · FindStat St001981

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

live ↗

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

Known value · FindStat St001981

About

Examples	4
Crosswalk	FindStat