AscentCountIncludingLargestEntry

Entries whose successor is in the same or a higher row, and the largest entry.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000507via StandardTableau✓↗

AscentCountIncludingLargestEntry(tableau)Entries whose successor is in the same or a higher row, and the largest entry.

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 size less the descents of the tableau; FindStat's St000507 counts the largest entry, which has no successor, as an ascent. The empty tableau gives 0.

Definition

Epsil, as the implementation runs it.

on StandardTableau, over _x

0 if Length(_x) < 1 else 1 +
                         (0 if Length(Flatten(_x)) < 2 else Apply((reading: list<integer>) => Apply(
                                                                                                     (lengths: list<integer>) => Apply(
                                                                                                                                        (rowIds: list<integer>) => Max(Sum(Map(Function(1 if rowIds[IndexOf(reading, e + 1)] <=
                                                                                                                                                                                             rowIds[IndexOf(reading, e)] else 0, e), Range(1, Length(reading) - 1, 1))), 0),
                                                                                                                                        Flatten(Map(
                                                                                                                                                     Function(Map(Function(i, j), Range(1, lengths[i], 1)), i),
                                                                                                                                                     Range(1, Length(_x)),
                                                                                                                                                   )),
                                                                                                                                      ),
                                                                                                     Map(Function(Length(r), r), _x),
                                                                                                   ), Flatten(_x)))

Examples

Basic #

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

live ↗

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

Properties #

FindStat's value for [[1]]

live ↗

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

Known value · FindStat St000507

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

live ↗

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

Known value · FindStat St000507

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

live ↗

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

Known value · FindStat St000507

About

Examples	4
Crosswalk	FindStat