MainDiagonalSum

The sum of the entries on the main diagonal, its trace.

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MainDiagonalSum(tableau)The sum of the entries on the main diagonal, its trace.

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 St000894. The entries are -1, 0 and 1, so the trace is the number of 1s on the diagonal less the number of -1s.

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(1 if cells[(i - 1) *
                                                                                                                   side +
                                                                                                                   i] ==
                                                                                                             1 else 0, i),
                                                                                               Range(1, Length(_x)),
                                                                                             )), 0) -
                                                                                  Max(Sum(Map(
                                                                                               Function(1 if cells[(i - 1) *
                                                                                                                   side +
                                                                                                                   i] ==
                                                                                                             -1 else 0, i),
                                                                                               Range(1, Length(_x)),
                                                                                             )), 0), Length(_x)), Flatten(_x))

Examples

Basic #

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

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St000894

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

live ↗

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

Known value · FindStat St000894

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

live ↗

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

Known value · FindStat St000894

About

Examples	4
Crosswalk	FindStat