TwelveTimesMajorIndexVarianceOfStandardTableaux

Twelve times the variance of the major index over the standard tableaux: the sum of the first n squares, less the sum of the squared hook lengths.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001178via IntegerPartition✓↗

TwelveTimesMajorIndexVarianceOfStandardTableaux(partition)Twelve times the variance of the major index of the standard tableaux: the sum of the first n squares, less the sum of the squares of the hook lengths.

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.
• Follows FindStat's description, whose Sage code does not parse; the values agree with it.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

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

Examples

Basic #

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

live ↗

In$\mathrm{TwelveTimesMajorIndexVarianceOfStandardTableaux}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$72$

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

live ↗

In$\mathrm{TwelveTimesMajorIndexVarianceOfStandardTableaux}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$80$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001178

FindStat's value for [2]

live ↗

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

Known value · FindStat St001178

FindStat's value for [3]

live ↗

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

Known value · FindStat St001178

About

Examples	5
Crosswalk	FindStat