FirstColumnOnePosition

The row of the unique 1 in the first column.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000193via AlternatingSignMatrix✓↗

FirstColumnOnePosition(tableau)The row of the unique 1 in the first column.

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.

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(i if cells[(i - 1) *
                                                                                                                    side +
                                                                                                                    1] ==
                                                                                                              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$\mathrm{FirstColumnOnePosition}({\operatorname{AlternatingSignMatrix}}([[0, 1, 0], [1, -1, 1], [0, 1, 0]]))$Out$2$

Properties #

FindStat's value for [[1]]

live ↗

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

Known value · FindStat St000193

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

live ↗

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

Known value · FindStat St000193

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

live ↗

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

Known value · FindStat St000193

About

Examples	4
Crosswalk	FindStat