CylindricalTableauOverlap

The number of rows, from the top, as long as the first row and starting, from the bottom, above the last entry of the first row.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001803via StandardTableau✓↗

CylindricalTableauOverlap(tableau)The number of rows, from the top, as long as the first row and starting, from the bottom, above the last entry of the first row.

Typed signatures

Type	Package	Applies
(standard_tableau) -> number	combinatorics	always

Domain: Tableau statistics

Details

• Defined over StandardTableau as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a StandardTableau 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.
• The overlap of the cylindrical tableau built from two copies (FindStat's St001803). The empty tableau gives 0.

Definition

Epsil, as the implementation runs it.

on StandardTableau, over _x

0 if Length(_x) < 1 else Apply((reading: list<integer>) => Apply(
                                                                  (lengths: list<integer>) => Apply(
                                                                                                     (firsts: list<integer>) => Max(Sum(Map(Function(1 if lengths[i] == lengths[1] &&
                                                                                                                                                          reading[lengths[1]] <
                                                                                                                                                          firsts[Length(_x) + 1 -
                                                                                                                                                                 i] else 0, i), Range(1, Length(_x)))), 0),
                                                                                                     Map(Function(r[1], r), _x),
                                                                                                   ),
                                                                  Map(
                                                                       Function(Length(r), r),
                                                                       _x,
                                                                     ),
                                                                ), Flatten(_x))

Examples

Basic #

the standard tableau $[[1,2,4],[3,6],[5]]$

live ↗

In$\mathrm{CylindricalTableauOverlap}({\operatorname{StandardTableau}}([[1, 2, 4], [3, 6], [5]]))$Out$1$

Properties #

FindStat's value for [[1]]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{StandardTableau}}([[1]]), \text{CylindricalTableauOverlap})$Out$0$

Known value · FindStat St001803

FindStat's value for [[1,2]]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{StandardTableau}}([[1, 2]]), \text{CylindricalTableauOverlap})$Out$0$

Known value · FindStat St001803

FindStat's value for [[1,2,3]]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{StandardTableau}}([[1, 2, 3]]), \text{CylindricalTableauOverlap})$Out$0$

Known value · FindStat St001803

About

Examples	4
Crosswalk	FindStat