TwiceMeanMajorIndexOfStandardTableaux

Twice the mean major index over the standard tableaux: n choose 2, less the sum of the row lengths choose 2, plus the sum of the column lengths choose 2.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001177via IntegerPartition✓↗

TwiceMeanMajorIndexOfStandardTableaux(partition)Twice the mean major index of the standard tableaux: n choose 2, less the sum over the rows of the row length choose 2, plus the sum over the columns of the column length choose 2.

Typed signatures

Type	Package	Applies
(integer_partition) -> number	combinatorics	always

Domain: Partition statistics

Details

• Defined over IntegerPartition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a IntegerPartition 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.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

0 if Length(_x) < 1 else Sum(_x) * (Sum(_x) - 1) / 2 -
                         Sum(Map(Function(i * (i - 1) / 2, i), _x)) + Sum(Map(
                                     Function(i * (i - 1) / 2, i),
                                     Map(
                                          Function(Count(Filter(
                                                                 Range(
                                                                        1,
                                                                        Length(_x),
                                                                      ),
                                                                 Function(_x[j] >=
                                                                          i, j),
                                                               )), i),
                                          Range(1, Max(_x)),
                                        ),
                                   ))

Examples

Basic #

the partition $4 + 2 + 1$ of $7$

live ↗

In$\mathrm{TwiceMeanMajorIndexOfStandardTableaux}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$18$

the partition $3 + 3 + 1$ of $7$

live ↗

In$\mathrm{TwiceMeanMajorIndexOfStandardTableaux}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$20$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001177

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([2]), \text{TwiceMeanMajorIndexOfStandardTableaux})$Out$0$

Known value · FindStat St001177

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{TwiceMeanMajorIndexOfStandardTableaux})$Out$0$

Known value · FindStat St001177

About

Examples	5
Crosswalk	FindStat