PlancherelCount

The square of the number of standard tableaux: the permutations whose Robinson-Schensted shape is this partition.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000284via IntegerPartition✓↗

PlancherelCount(partition)The square of the number of standard tableaux: the permutations whose Robinson-Schensted shape is this partition.

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

1 if Length(_x) < 1 else ((1 if Sum(_x) == 0 else Product(Range(1, Sum(_x)))) /
                          Product(Map(Function(Product(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, j),
                                                            Range(1, _x[i]),
                                                          )), i), Range(
                                                                        1,
                                                                        Length(_x),
                                                                      )))) ^ 2

Examples

Basic #

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

live ↗

In$\mathrm{PlancherelCount}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$1\,225$

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

live ↗

In$\mathrm{PlancherelCount}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$441$

Properties #

FindStat's value for [2]

live ↗

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

Known value · FindStat St000284

FindStat's value for [3]

live ↗

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

Known value · FindStat St000284

FindStat's value for [4]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([4]), \text{PlancherelCount})$Out$1$

Known value · FindStat St000284

About

Examples	5
Crosswalk	FindStat