PathLengthToLargestEntry

The length of the path from the top left cell to the largest entry, one step per cell down or to the right.

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PathLengthToLargestEntry(tableau)The length of the path from the top left cell to the largest entry, one step per cell down or to the right.

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.
• FindStat sums the 0-based row and column of the cell. 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(
                                (lasts: list<integer>) => Apply(
                                                                 (lengths: list<integer>) => Apply(
                                                                                                    (largestRow: integer) => largestRow + lengths[largestRow] -
                                                                                                                             2,
                                                                                                    IndexOf(lasts, Max(lasts)),
                                                                                                  ),
                                                                 Map(
                                                                      Function(Length(r), r),
                                                                      _x,
                                                                    ),
                                                               ),
                                Map(Function(Last(r), r), _x),
                              )

Examples

Basic #

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

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St000744

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

live ↗

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

Known value · FindStat St000744

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

live ↗

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

Known value · FindStat St000744

About

Examples	4
Crosswalk	FindStat