ShiftedNaturalComajorIndex

The sum of n - e over the natural descents e, plus the sum of (r - 1) times the length of row r.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001697via StandardTableau✓↗

ShiftedNaturalComajorIndex(tableau)The sum of n - e over the natural descents e, plus the sum of (r - 1) times the length of row r.

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 natural comajor index shifted by the weighted size of the shape (FindStat's St001697). A tableau of fewer than two entries has no natural descent and a weighted size of 0.

Definition

Epsil, as the implementation runs it.

on StandardTableau, over _x

0 if Length(Flatten(_x)) < 2 else Apply((reading: list<integer>) => Apply(
                                                                           (lengths: list<integer>) => Apply(
                                                                                                              (rowIds: list<integer>) => Max(Sum(Map(
                                                                                                                                                      Function(Length(reading) - e if rowIds[IndexOf(reading, e + 1)] <
                                                                                                                                                                                      rowIds[IndexOf(reading, e)] else 0, e),
                                                                                                                                                      Range(1, Length(reading) - 1, 1),
                                                                                                                                                    )), 0) +
                                                                                                                                         Max(Sum(Map(Function((i - 1) *
                                                                                                                                                              lengths[i], i), Range(1, Length(_x)))), 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$\mathrm{ShiftedNaturalComajorIndex}({\operatorname{StandardTableau}}([[1, 2, 4], [3, 6], [5]]))$Out$8$

Properties #

FindStat's value for [[1]]

live ↗

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

Known value · FindStat St001697

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

live ↗

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

Known value · FindStat St001697

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

live ↗

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

Known value · FindStat St001697

About

Examples	4
Crosswalk	FindStat