ShynarInversionCount

Pairs of entries e < f with f strictly southwest of e.

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ShynarInversionCount(tableau)Pairs of entries e < f with f strictly southwest of e.

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.
• Shynar's inversion number (FindStat's St000057). A tableau of fewer than two entries gives 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>) => Apply(
                                                                                                                                                (columnIds: list<integer>) => Apply(
                                                                                                                                                                                     (entryRows: list<integer>) => Apply(
                                                                                                                                                                                                                          (entryColumns: list<integer>) => Max(Sum(Map(
                                                                                                                                                                                                                                                                        Function(Max(Sum(Map(Function(1 if entryRows[f] > entryRows[e] &&
                                                                                                                                                                                                                                                                                                           entryColumns[f] < entryColumns[e] else 0, f), Range(e + 1, Length(reading), 1))), 0), e),
                                                                                                                                                                                                                                                                        Range(1, Length(reading) - 1, 1),
                                                                                                                                                                                                                                                                      )), 0),
                                                                                                                                                                                                                          Map(Function(columnIds[IndexOf(reading, e)], e), Range(1, Length(reading), 1)),
                                                                                                                                                                                                                        ),
                                                                                                                                                                                     Map(Function(rowIds[IndexOf(reading, e)], e), Range(1, Length(reading), 1)),
                                                                                                                                                                                   ),
                                                                                                                                                Flatten(Map(Function(Range(1, lengths[i], 1), i), Range(1, Length(_x)))),
                                                                                                                                              ),
                                                                                                              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$\mathrm{ShynarInversionCount}({\operatorname{StandardTableau}}([[1, 2, 4], [3, 6], [5]]))$Out$4$

Properties #

FindStat's value for [[1]]

live ↗

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

Known value · FindStat St000057

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

live ↗

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

Known value · FindStat St000057

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

live ↗

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

Known value · FindStat St000057

About

Examples	4
Crosswalk	FindStat