FirstRowMissingBoxCount

The boxes of the first row that the shape leaves out: the first part of the inner partition.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001435via SkewPartition✓↗

FirstRowMissingBoxCount(shape)The boxes of the first row that the shape leaves out: the first part of the inner partition.

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 St001435. 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>) => mu[1],
                                                                                                 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$\mathrm{FirstRowMissingBoxCount}({\operatorname{SkewPartition}}([[3, 1], [1]]))$Out$1$

Properties #

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

live ↗

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

Known value · FindStat St001435

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

live ↗

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

Known value · FindStat St001435

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

live ↗

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

Known value · FindStat St001435

About

Examples	4
Crosswalk	FindStat