MajorIndexMinusWeightedSize

The major index of the tableau less the weighted size of its shape.

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MajorIndexMinusWeightedSize(tableau)The major index of the tableau less the weighted size of its shape.

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.
• FindStat's St001699. The major index is the sum of the descents, the entries whose successor is in a strictly lower row; the weighted size is the sum over the rows of the row's number counted from 0 times its length. A tableau of fewer than two entries has neither.

Definition

Epsil, as the implementation runs it.

on StandardTableau, over _x

0 if Length(Flatten(_x)) < 2 else Apply((reading: list<integer>) => Apply(
                                                                           (lengths: list<integer>) => Apply(
                                                                                                              (rowIds: list<integer>) => Max(Sum(Map(Function(e if rowIds[IndexOf(reading, e + 1)] >
                                                                                                                                                                   rowIds[IndexOf(reading, e)] else 0, e), Range(1, Length(reading) - 1, 1))), 0) -
                                                                                                                                         Max(Sum(Map(Function((i - 1) *
                                                                                                                                                              lengths[i], i), Range(1, Length(_x)))), 0),
                                                                                                              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$\mathrm{MajorIndexMinusWeightedSize}({\operatorname{StandardTableau}}([[1, 2, 4], [3, 6], [5]]))$Out$2$

Properties #

FindStat's value for [[1]]

live ↗

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

Known value · FindStat St001699

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

live ↗

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

Known value · FindStat St001699

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

live ↗

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

Known value · FindStat St001699

About

Examples	4
Crosswalk	FindStat