SkewConnectedComponentCount

The connected components of the shape: rows that start a component, the first row and every row that shares no column with the row above.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001490via SkewPartition✓↗

SkewConnectedComponentCount(shape)The connected components of the shape: rows that start a component, the first row and every row that shares no column with the row above.

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 St001490, the components of the cell poset. Row i meets the row above in a column exactly when λᵢ exceeds μᵢ₋₁. 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((1 if i == 1 else 1 if lam[i] <=
                                                                                                                                                                    mu[i - 1] else 0) if lam[i] > mu[i] else 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$\mathrm{SkewConnectedComponentCount}({\operatorname{SkewPartition}}([[3, 1], [1]]))$Out$2$

Properties #

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

live ↗

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

Known value · FindStat St001490

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SkewPartition}}([[2], []]), \text{SkewConnectedComponentCount})$Out$1$

Known value · FindStat St001490

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SkewPartition}}([[3], []]), \text{SkewConnectedComponentCount})$Out$1$

Known value · FindStat St001490

About

Examples	4
Crosswalk	FindStat