BorderSkewStandardTableauCount

Standard tableaux of the skew shape that adds a cell to every column of the diagram: the size factorial over the factorials of the cells added to each row (the size less the largest part, then the gaps between consecutive parts).

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001387via IntegerPartition✓↗

BorderSkewStandardTableauCount(partition)The standard tableaux of the skew shape that adds one cell to every column of the diagram, and one in each new column to the right: the size factorial over the factorials of the cells added to each row, which are the first row's size less the largest part, then the differences between consecutive 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 (1 if Sum(_x) == 0 else Product(Range(1, Sum(_x)))) /
                         ((1 if Sum(_x) - Max(_x) == 0 else Product(Range(
                                                                           1,
                                                                           Sum(_x) -
                                                                           Max(_x),
                                                                         ))) *
                          Product(Map(Function(1 if _x[i] - (0 if i ==
                                                                  Length(_x) else _x[i + 1]) ==
                                                    0 else Product(Range(
                                                                         1,
                                                                         _x[i] -
                                                                         (0 if i ==
                                                                               Length(_x) else _x[i + 1]),
                                                                       )), i), Range(1, Length(_x)))))

Examples

Basic #

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

live ↗

In$\mathrm{BorderSkewStandardTableauCount}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$420$

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

live ↗

In$\mathrm{BorderSkewStandardTableauCount}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$105$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001387

FindStat's value for [2]

live ↗

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

Known value · FindStat St001387

FindStat's value for [3]

live ↗

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

Known value · FindStat St001387

About

Examples	5
Crosswalk	FindStat