TwoByTwoSquareCount

The cells of the shape whose neighbors to the right, below and below to the right are also in it.

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TwoByTwoSquareCount(shape)The cells of the shape whose neighbors to the right, below and below to the right are also in it.

Typed signatures

Type	Package	Applies
(skew_partition) -> number	combinatorics	always

Domain: Skew partition statistics

Details

• Defined over SkewPartition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a SkewPartition 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 St001596. A cell (i, j) with the cell (i + 1, j + 1) also in the shape is the top left of a square, so row i counts the columns from μᵢ to λᵢ₊₁ - 2. The empty shape gives 0.

Definition

Epsil, as the implementation runs it.

on SkewPartition, over _x

0 if Length(_x[1]) == 0 else Apply((lam: list<integer>) => Apply((raw: list<integer>) => Apply(
                                                                                                (mu: list<integer>) => Max(Sum(Map(
                                                                                                                                    Function(0 if i == Length(lam) else Max(lam[i + 1] -
                                                                                                                                                                            1 -
                                                                                                                                                                            mu[i], 0), i),
                                                                                                                                    Range(1, Length(lam)),
                                                                                                                                  )), 0),
                                                                                                Map(
                                                                                                     Function(raw[i] if i <= Length(raw) else 0, i),
                                                                                                     Range(1, Length(lam)),
                                                                                                   ),
                                                                                              ), _x[2]), _x[1])

Examples

Basic #

the skew partition $[[3,1],[1]]$

live ↗

In$\mathrm{TwoByTwoSquareCount}({\operatorname{SkewPartition}}([[3, 1], [1]]))$Out$0$

Properties #

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

live ↗

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

Known value · FindStat St001596

FindStat's value for [[2],[]]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SkewPartition}}([[2], []]), \text{TwoByTwoSquareCount})$Out$0$

Known value · FindStat St001596

FindStat's value for [[3],[]]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SkewPartition}}([[3], []]), \text{TwoByTwoSquareCount})$Out$0$

Known value · FindStat St001596

About

Examples	4
Crosswalk	FindStat