MahonianWordExcedances

Excedances of the Mahonian word.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001839via SetPartition✓↗

MahonianWordExcedances(partition)Excedances of the Mahonian word.

Typed signatures

Type	Package	Applies
(set_partition) -> number	combinatorics	always

Domain: Set partition statistics

Details

• Defined over SetPartition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a SetPartition 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.

Definition

Epsil, as the implementation runs it.

on SetPartition, over _x

0 if Length(_x) < 1 else Count(Filter(
                                       Range(1, Length(Flatten(_x))),
                                       Function(Count(Filter(
                                                              Map(Function(
                                                                            Last(b),
                                                                            b,
                                                                          ), _x),
                                                              Function(c <
                                                                       Sum(Map(
                                                                                Function(
                                                                                          Last(_x[q]) if Contains(_x[q], i) else 0,
                                                                                          q,
                                                                                        ),
                                                                                Range(1, Length(_x)),
                                                                              )), c),
                                                            )) > Count(Filter(
                                                                              Range(1, Length(_x)),
                                                                              Function(Sum(Map(
                                                                                                Function(Length(_x[c]) if Map(Function(Last(b), b), _x)[c] <=
                                                                                                                          Map(Function(Last(b), b), _x)[k] else 0, c),
                                                                                                Range(1, Length(_x)),
                                                                                              )) <
                                                                                       i, k),
                                                                            )), i),
                                     ))

Examples

Basic #

the partition $\{1,3\} \mid \{2\}$

live ↗

In$\mathrm{MahonianWordExcedances}({\operatorname{SetPartition}}([[1, 3], [2]]))$Out$1$

the partition $\{1\} \mid \{2,4\} \mid \{3\}$

live ↗

In$\mathrm{MahonianWordExcedances}({\operatorname{SetPartition}}([[1], [2, 4], [3]]))$Out$1$

Properties #

FindStat's value for {{1}}

live ↗

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

Known value · FindStat St001839

FindStat's value for {{1,2}}

live ↗

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

Known value · FindStat St001839

FindStat's value for {{1,2,3}}

live ↗

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

Known value · FindStat St001839

About

Examples	5
Crosswalk	FindStat