SemistandardTableauCountWithEntriesUpToLength

Semistandard tableaux of the shape with entries at most the number of parts.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000704via IntegerPartition✓↗

SemistandardTableauCountWithEntriesUpToLength(partition)The number of semistandard tableaux of the shape with entries at most the number of parts.

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(Length(_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$\mathrm{SemistandardTableauCountWithEntriesUpToLength}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$15$

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

live ↗

In$\mathrm{SemistandardTableauCountWithEntriesUpToLength}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$6$

Properties #

FindStat's value for [2]

live ↗

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

Known value · FindStat St000704

FindStat's value for [3]

live ↗

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

Known value · FindStat St000704

FindStat's value for [4]

live ↗

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

Known value · FindStat St000704

About

Examples	5
Crosswalk	FindStat