DualImmaculateMonomialColumnSum

The sum of the entries in the column of the composition in the matrix from the dual immaculate to the monomial quasisymmetric functions.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000817via IntegerComposition✓↗

DualImmaculateMonomialColumnSum(composition)The sum of the entries in the column of the composition in the matrix from the dual immaculate to the monomial quasisymmetric functions.

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.
• FindStat's St000817: the number of immaculate tableaux of the shape with any gapless content (Berg, Bergeron, Saliola, Serrano and Zabrocki).

Definition

Epsil, as the implementation runs it.

on IntegerComposition, over _x

0 if Length(_x) < 1 else Max(Sum(Map(Function(Sum(Fold(
                                                        Function(Map(
                                                                      Function(0 if p == 1 else Binomial(m - p +
                                                                                                         b, b - 1) *
                                                                                                Sum(Map(
                                                                                                         Function(a[q], q),
                                                                                                         Range(1, Max(p - 1, 1)),
                                                                                                       )), p),
                                                                      Range(
                                                                             1,
                                                                             m +
                                                                             1,
                                                                           ),
                                                                    ), a, b),
                                                        Map(Function(
                                                                      1 if p ==
                                                                           1 else 0,
                                                                      p,
                                                                    ), Range(
                                                                             1,
                                                                             m +
                                                                             1,
                                                                           )),
                                                        _x,
                                                      )) * Sum(Map(
                                                                   Function((-1) ^
                                                                            (k -
                                                                             m) *
                                                                            Binomial(k, m), k),
                                                                   Range(
                                                                          m,
                                                                          Sum(_x),
                                                                        ),
                                                                 )), m), Range(1, Sum(_x)))), 0)

Examples

Basic #

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

live ↗

In$\mathrm{DualImmaculateMonomialColumnSum}({\operatorname{IntegerComposition}}([2, 1, 3, 1, 1]))$Out$288$

Properties #

FindStat's value for [2]

live ↗

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

Known value · FindStat St000817

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([3]), \text{DualImmaculateMonomialColumnSum})$Out$4$

Known value · FindStat St000817

FindStat's value for [4]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([4]), \text{DualImmaculateMonomialColumnSum})$Out$8$

Known value · FindStat St000817

About

Examples	4
Crosswalk	FindStat