MahonianWordZIndex

The major indices of the Mahonian word restricted to each pair of letters, summed.

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MahonianWordZIndex(partition)The major indices of the Mahonian word restricted to each pair of letters, summed.

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) < 2 else Sum(Map(Function(Sum(Map(Function(0 if Length(Filter(Map(
                                                                                   Function(Count(Filter(
                                                                                                          Map(Function(Last(b), b), _x),
                                                                                                          Function(c <
                                                                                                                   Sum(Map(
                                                                                                                            Function(Last(_x[q]) if Contains(_x[q], e) else 0, q),
                                                                                                                            Range(1, Length(_x)),
                                                                                                                          )), c),
                                                                                                        )), e),
                                                                                   Range(1, Length(Flatten(_x))),
                                                                                 ), Function(v == r ||
                                                                                            v == s, v))) <
                                                                2 else Sum(Map(
                                                                               Function(i if Filter(
                                                                                                     Map(Function(Count(Filter(
                                                                                                                                Map(Function(Last(b), b), _x),
                                                                                                                                Function(c <
                                                                                                                                         Sum(Map(
                                                                                                                                                  Function(Last(_x[q]) if Contains(_x[q], e) else 0, q),
                                                                                                                                                  Range(1, Length(_x)),
                                                                                                                                                )), c),
                                                                                                                              )), e), Range(1, Length(Flatten(_x)))),
                                                                                                     Function(v == r ||
                                                                                                              v == s, v),
                                                                                                   )[i] >
                                                                                             Filter(
                                                                                                     Map(Function(Count(Filter(
                                                                                                                                Map(Function(Last(b), b), _x),
                                                                                                                                Function(c <
                                                                                                                                         Sum(Map(
                                                                                                                                                  Function(Last(_x[q]) if Contains(_x[q], e) else 0, q),
                                                                                                                                                  Range(1, Length(_x)),
                                                                                                                                                )), c),
                                                                                                                              )), e), Range(1, Length(Flatten(_x)))),
                                                                                                     Function(v == r ||
                                                                                                              v == s, v),
                                                                                                   )[i + 1] else 0, i),
                                                                               Range(
                                                                                      1,
                                                                                      Length(Filter(
                                                                                                     Map(Function(Count(Filter(
                                                                                                                                Map(Function(Last(b), b), _x),
                                                                                                                                Function(c <
                                                                                                                                         Sum(Map(
                                                                                                                                                  Function(Last(_x[q]) if Contains(_x[q], e) else 0, q),
                                                                                                                                                  Range(1, Length(_x)),
                                                                                                                                                )), c),
                                                                                                                              )), e), Range(1, Length(Flatten(_x)))),
                                                                                                     Function(v == r ||
                                                                                                              v == s, v),
                                                                                                   )) -
                                                                                      1,
                                                                                    ),
                                                                             )), s), Range(r + 1, Length(_x) - 1))), r), Range(0, Length(_x) - 2)))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for {{1}}

live ↗

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

Known value · FindStat St001843

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

live ↗

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

Known value · FindStat St001843

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

live ↗

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

Known value · FindStat St001843

About

Examples	5
Crosswalk	FindStat