MaximalDiagonalSum

The largest sum of the entries along a diagonal.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000888via AlternatingSignMatrix✓↗

MaximalDiagonalSum(tableau)The largest sum of the entries along a diagonal.

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.
• The diagonals are the cells (i, j) with i - j fixed. A sum is floored at 0, which no sum of a corner cell falls below, so the largest is unchanged.

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(Map(
                                                                                           Function(Max(Sum(Map(
                                                                                                                 Function(cells[(i - 1) *
                                                                                                                                side +
                                                                                                                                (i - d)] if i - d >=
                                                                                                                                     1 &&
                                                                                                                                     i - d <=
                                                                                                                                     side else 0, i),
                                                                                                                 Range(1, Length(_x)),
                                                                                                               )), 0), d),
                                                                                           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$\mathrm{MaximalDiagonalSum}({\operatorname{AlternatingSignMatrix}}([[0, 1, 0], [1, -1, 1], [0, 1, 0]]))$Out$2$

Properties #

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

live ↗

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

Known value · FindStat St000888

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

live ↗

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

Known value · FindStat St000888

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

live ↗

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

Known value · FindStat St000888

About

Examples	4
Crosswalk	FindStat