LastEntryOnMainDiagonal

The last entry on the main diagonal.

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LastEntryOnMainDiagonal(tableau)The last entry on the main diagonal.

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 empty tableau, which has no such entry, 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(
                                                                                                     (diagonal: integer) => reading[Max(Sum(Map(
                                                                                                                                                 Function(lengths[i] if i < diagonal else 0, i),
                                                                                                                                                 Range(1, Length(_x)),
                                                                                                                                               )), 0) +
                                                                                                                                    diagonal],
                                                                                                     Max(Sum(Map(
                                                                                                                  Function(1 if lengths[i] >= i else 0, i),
                                                                                                                  Range(1, Length(_x)),
                                                                                                                )), 0),
                                                                                                   ),
                                                                  Map(
                                                                       Function(Length(r), r),
                                                                       _x,
                                                                     ),
                                                                ), Flatten(_x))

Examples

Basic #

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

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St000735

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

live ↗

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

Known value · FindStat St000735

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

live ↗

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

Known value · FindStat St000735

About

Examples	4
Crosswalk	FindStat