DistinctDiagonalSumCount

The number of distinct sums of the entries along a diagonal.

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DistinctDiagonalSumCount(tableau)The number of distinct sums 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.
• FindStat's St000893. A middle diagonal can sum below 0, so each sum is raised by the side before it is floored; raising every sum alike leaves the number of distinct ones.

Definition

Epsil, as the implementation runs it.

on AlternatingSignMatrix, over _x

0 if Length(_x) < 1 else Apply((cells: list<integer>) => Apply((side: integer) => Length(Unique(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)))) +
                                                                                                                  side, 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$\mathrm{DistinctDiagonalSumCount}({\operatorname{AlternatingSignMatrix}}([[0, 1, 0], [1, -1, 1], [0, 1, 0]]))$Out$3$

Properties #

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

live ↗

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

Known value · FindStat St000893

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

live ↗

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

Known value · FindStat St000893

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

live ↗

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

Known value · FindStat St000893

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Examples	4
Crosswalk	FindStat