AttackingPairCount

Pairs of cells of the shape with one strictly west of the other in a row, or the first in the row below and strictly east of the second.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000016via StandardTableau✓↗

AttackingPairCount(tableau)Pairs of cells of the shape with one strictly west of the other in a row, or the first in the row below and strictly east of the second.

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.
• A cell with k cells west of it in its row makes k pairs there and k more with the row above, if it has a row above. It depends on the shape alone (FindStat's St000016). 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) *
                                                                                                                                                                                          (2 if rowIds[i] > 1 else 1), 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$\mathrm{AttackingPairCount}({\operatorname{StandardTableau}}([[1, 2, 4], [3, 6], [5]]))$Out$5$

Properties #

FindStat's value for [[1]]

live ↗

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

Known value · FindStat St000016

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

live ↗

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

Known value · FindStat St000016

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

live ↗

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

Known value · FindStat St000016

About

Examples	4
Crosswalk	FindStat