StandardRibbonTableauCount

Standard fillings of the ribbon whose rows have these lengths: the permutations of 1 .. n with these partial sums as their descents.

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StandardRibbonTableauCount(composition)Standard fillings of the ribbon whose rows have these lengths: the permutations of 1 .. n with these partial sums as their descents.

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.
• Stanley, Enumerative Combinatorics, volume 1, section 1.4: Proposition 1.4.1 counts permutations whose descents lie in a set, and inclusion-exclusion over it gives this count. FindStat cites the formula as section 14.4.

Definition

Epsil, as the implementation runs it.

on IntegerComposition, over _x

1 if Length(_x) < 1 else Last(Fold(Function(Join(
                                                  a,
                                                  [Sum(Map(Function((-1) ^ (b -
                                                                            j) *
                                                                    a[j] /
                                                                    Product(Range(
                                                                                   1,
                                                                                   Sum(Map(
                                                                                            Function(_x[u], u),
                                                                                            Filter(
                                                                                                    Range(1, Length(_x)),
                                                                                                    Function(u <
                                                                                                             b + 1, u),
                                                                                                  ),
                                                                                          )) -
                                                                                   Sum(Map(
                                                                                            Function(_x[u], u),
                                                                                            Filter(
                                                                                                    Range(1, Length(_x)),
                                                                                                    Function(u <
                                                                                                             j - 1 +
                                                                                                             1, u),
                                                                                                  ),
                                                                                          )),
                                                                                 )), j), Range(1, b)))],
                                                ), a, b), [Product(Range(
                                                                         1,
                                                                         Sum(_x),
                                                                       ))], Range(1, Length(_x))))

Examples

Basic #

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

live ↗

In$\mathrm{StandardRibbonTableauCount}({\operatorname{IntegerComposition}}([2, 1, 3, 1, 1]))$Out$413$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000277

FindStat's value for [2]

live ↗

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

Known value · FindStat St000277

FindStat's value for [3]

live ↗

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

Known value · FindStat St000277

About

Examples	4
Crosswalk	FindStat