ModularMajorIndex

The major index modulo the number of entries.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001485via BinaryWord✓↗St000693via StandardTableau✓↗

ModularMajorIndex(tableau)The major index modulo the number of entries.

Typed signatures

Type	Package	Applies
(binary_word | 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.
• The major index is the sum of the descents, the entries whose successor is in a strictly lower row. A tableau of fewer than two entries gives 0.

Definition

Epsil, as the implementation runs it.

on BinaryWord, over _x

0 if Length(_x) < 1 else (0 if Length(_x) < 2 else Sum(Filter(
                                                               Range(
                                                                      1,
                                                                      Length(_x) -
                                                                      1,
                                                                    ),
                                                               Function(_x[i] >
                                                                        _x[i + 1], i),
                                                             ))) % Length(_x)

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) %
                                                                                                                                         Length(reading),
                                                                                                              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$\mathrm{ModularMajorIndex}({\operatorname{StandardTableau}}([[1, 2, 4], [3, 6], [5]]))$Out$0$

Properties #

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

live ↗

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

Known value · FindStat St000693

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

live ↗

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

Known value · FindStat St000693

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

live ↗

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

Known value · FindStat St000693

About

Examples	4
Crosswalk	FindStat ×2