AttackingInversionCount

Attacking pairs of cells whose entries are out of order: the cell in the lower row, strictly east of the other, holds the larger entry.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000017via StandardTableau✓↗

AttackingInversionCount(tableau)Attacking pairs of cells whose entries are out of order: the cell in the lower row, strictly east of the other, holds the larger entry.

Typed signatures

Type	Package	Applies
(standard_tableau) -> number	combinatorics	always

Domain: Tableau statistics

Details

• Defined over StandardTableau as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a StandardTableau 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.
• Pairs in a row are in order. Across rows, the lower cell's entry exceeds the entry above it, which exceeds the entry of the cell it attacks, so every such pair is an inversion (FindStat's St000017). The empty tableau gives 0.

Definition

Epsil, as the implementation runs it.

on StandardTableau, over _x

0 if Length(_x) < 1 else Apply((reading: list<integer>) => Apply(
                                                                  (lengths: list<integer>) => Apply(
                                                                                                     (rowIds: list<integer>) => Apply(
                                                                                                                                       (columnIds: list<integer>) => Max(Sum(Map(Function(columnIds[i] - 1 if rowIds[i] > 1 else 0, i), Range(1, Length(reading), 1))), 0),
                                                                                                                                       Flatten(Map(Function(Range(1, lengths[i], 1), i), Range(1, Length(_x)))),
                                                                                                                                     ),
                                                                                                     Flatten(Map(
                                                                                                                  Function(Map(Function(i, j), Range(1, lengths[i], 1)), i),
                                                                                                                  Range(1, Length(_x)),
                                                                                                                )),
                                                                                                   ),
                                                                  Map(
                                                                       Function(Length(r), r),
                                                                       _x,
                                                                     ),
                                                                ), Flatten(_x))

Examples

Basic #

the standard tableau $[[1,2,4],[3,6],[5]]$

live ↗

In$\mathrm{AttackingInversionCount}({\operatorname{StandardTableau}}([[1, 2, 4], [3, 6], [5]]))$Out$1$

Properties #

FindStat's value for [[1]]

live ↗

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

Known value · FindStat St000017

FindStat's value for [[1,2]]

live ↗

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

Known value · FindStat St000017

FindStat's value for [[1,2,3]]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{StandardTableau}}([[1, 2, 3]]), \text{AttackingInversionCount})$Out$0$

Known value · FindStat St000017

About

Examples	4
Crosswalk	FindStat