MinimalCylindricalInnerHeight

The number of rows less the overlap of the cylindrical tableau.

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MinimalCylindricalInnerHeight(tableau)The number of rows less the overlap of the cylindrical tableau.

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 height of the rectangular inner shape of the cylindrical tableau built from two copies (FindStat's St001804). 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>) => Length(_x) -
                                                                                                                                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$\mathrm{MinimalCylindricalInnerHeight}({\operatorname{StandardTableau}}([[1, 2, 4], [3, 6], [5]]))$Out$2$

Properties #

FindStat's value for [[1]]

live ↗

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

Known value · FindStat St001804

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

live ↗

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

Known value · FindStat St001804

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

live ↗

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

Known value · FindStat St001804

About

Examples	4
Crosswalk	FindStat