MahonianWordInversions

Inversions of the Mahonian word.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001841via SetPartition✓↗

MahonianWordInversions(partition)Inversions 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(Flatten(_x)) < 2 else Sum(Map(Function(Count(Filter(
                                                                 Range(
                                                                        i + 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(
                                                                                        Map(Function(Last(b), b), _x),
                                                                                        Function(c <
                                                                                                 Sum(Map(
                                                                                                          Function(Last(_x[q]) if Contains(_x[q], j) else 0, q),
                                                                                                          Range(1, Length(_x)),
                                                                                                        )), c),
                                                                                      )), j),
                                                               )), i), Range(
                                                                             1,
                                                                             Length(Flatten(_x)) -
                                                                             1,
                                                                           )))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for {{1}}

live ↗

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

Known value · FindStat St001841

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

live ↗

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

Known value · FindStat St001841

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

live ↗

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

Known value · FindStat St001841

About

Examples	5
Crosswalk	FindStat