SemistandardTableauCountWithEntriesUpToSize

Semistandard tableaux of the shape with entries at most the size of the partition.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000705via IntegerPartition✓↗

SemistandardTableauCountWithEntriesUpToSize(partition)The number of semistandard tableaux of the shape with entries at most the size of the 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 Product(Map(Function(
                                               Product(Map(
                                                            Function(Sum(_x) +
                                                                     (j - i), j),
                                                            Range(1, _x[i]),
                                                          )),
                                               i,
                                             ), Range(1, Length(_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),
                                                                     )))

Examples

Basic #

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

live ↗

In$\mathrm{SemistandardTableauCountWithEntriesUpToSize}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$7\,350$

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

live ↗

In$\mathrm{SemistandardTableauCountWithEntriesUpToSize}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$3\,528$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000705

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([2]), \text{SemistandardTableauCountWithEntriesUpToSize})$Out$3$

Known value · FindStat St000705

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{SemistandardTableauCountWithEntriesUpToSize})$Out$10$

Known value · FindStat St000705

About

Examples	5
Crosswalk	FindStat