StandardImmaculateTableauCount

Standard immaculate tableaux of the shape: fillings of 1 .. n increasing along the rows and down the first column.

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StandardImmaculateTableauCount(composition)Standard immaculate tableaux of the shape: fillings of 1 .. n increasing along the rows and down the first column.

Typed signatures

Type	Package	Applies
(composition) -> number	combinatorics	always

Domain: Composition statistics

Details

• Defined over IntegerComposition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a IntegerComposition 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.
• Berg, Bergeron, Saliola, Serrano and Zabrocki, A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions (Proposition 3.13).

Definition

Epsil, as the implementation runs it.

on IntegerComposition, over _x

1 if Length(_x) < 1 else Product(Range(1, Sum(_x))) / Product(Map(
                                                                   Function((Sum(_x) -
                                                                             Sum(Map(
                                                                                      Function(_x[j], j),
                                                                                      Range(1, i),
                                                                                    )) +
                                                                             _x[i]) *
                                                                            (Product(Range(1, _x[i])) /
                                                                             _x[i]), i),
                                                                   Range(
                                                                          1,
                                                                          Length(_x),
                                                                        ),
                                                                 ))

Examples

Basic #

the composition $[2,1,3,1,1]$

live ↗

In$\mathrm{StandardImmaculateTableauCount}({\operatorname{IntegerComposition}}([2, 1, 3, 1, 1]))$Out$42$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000047

FindStat's value for [2]

live ↗

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

Known value · FindStat St000047

FindStat's value for [3]

live ↗

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

Known value · FindStat St000047

About

Examples	4
Crosswalk	FindStat