StandardTableauCountWithMajorIndexCongruentToOneModuloSize

The standard Young tableaux whose major index leaves remainder 1 on division by the size of the shape (never, when the size is 1).

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001440via IntegerPartition✓↗

StandardTableauCountWithMajorIndexCongruentToOneModuloSize(partition)The standard Young tableaux whose major index leaves remainder 1 on division by the size of the shape (never, when the size is 1).

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 Apply((m: number) => Apply(
                                                     (q: list<integer>) => Sum(Map(
                                                                                    Function(q[i] if (i - 1 +
                                                                                                      m) %
                                                                                                     Sum(_x) ==
                                                                                                     1 else 0, i),
                                                                                    Range(1, Sum(_x) *
                                                                                             (Sum(_x) + 1) /
                                                                                             2 +
                                                                                             1),
                                                                                  )),
                                                     Apply(
                                                            (c: list<integer>) => Fold(
                                                                                        Function(Map(
                                                                                                      Function(Sum(Map(
                                                                                                                        Function(a[i -
                                                                                                                                   t * b], t),
                                                                                                                        Range(0, (i - 1 -
                                                                                                                                  (i - 1) %
                                                                                                                                  b) /
                                                                                                                                 b),
                                                                                                                      )), i),
                                                                                                      Range(1, Sum(_x) *
                                                                                                               (Sum(_x) + 1) /
                                                                                                               2 +
                                                                                                               1),
                                                                                                    ), a, b),
                                                                                        Fold(
                                                                                              Function(Map(
                                                                                                            Function(a[i] -
                                                                                                                     a[i - b] if i > b else a[i], i),
                                                                                                            Range(1, Sum(_x) *
                                                                                                                     (Sum(_x) + 1) /
                                                                                                                     2 +
                                                                                                                     1),
                                                                                                          ), a, b),
                                                                                              Map(
                                                                                                   Function(1 if i == 1 else 0, i),
                                                                                                   Range(1, Sum(_x) *
                                                                                                            (Sum(_x) + 1) /
                                                                                                            2 +
                                                                                                            1),
                                                                                                 ),
                                                                                              Range(1, Sum(_x)),
                                                                                            ),
                                                                                        Flatten(Map(
                                                                                                     Function(Map(Function(_x[i] - j +
                                                                                                                           (c[j] - i) +
                                                                                                                           1, j), Range(1, _x[i])), i),
                                                                                                     Range(1, Length(_x)),
                                                                                                   )),
                                                                                      ),
                                                            Map(
                                                                 Function(Count(Filter(
                                                                                        Range(1, Length(_x)),
                                                                                        Function(_x[j] >= i, j),
                                                                                      )), i),
                                                                 Range(
                                                                        1,
                                                                        Max(_x),
                                                                      ),
                                                               ),
                                                          ),
                                                   ), Sum(Map(
                                                              Function((i - 1) *
                                                                       _x[i], i),
                                                              Range(
                                                                     1,
                                                                     Length(_x),
                                                                   ),
                                                            )))

Examples

Basic #

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

live ↗

In$\mathrm{StandardTableauCountWithMajorIndexCongruentToOneModuloSize}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$5$

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001440

FindStat's value for [2]

live ↗

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

Known value · FindStat St001440

FindStat's value for [3]

live ↗

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

Known value · FindStat St001440

About

Examples	5
Crosswalk	FindStat