MahonianWordMajorIndex

The major index of the Mahonian word.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001842via SetPartition✓↗

MahonianWordMajorIndex(partition)The major index 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(i if 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], i + 1) else 0,
                                                                                                 q,
                                                                                               ), Range(1, Length(_x)))), c),
                                                                    )) else 0, i), Range(
                                                                            1,
                                                                            Length(Flatten(_x)) -
                                                                            1,
                                                                          )))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for {{1}}

live ↗

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

Known value · FindStat St001842

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

live ↗

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

Known value · FindStat St001842

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

live ↗

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

Known value · FindStat St001842

About

Examples	5
Crosswalk	FindStat