MahonianWordDescents

Descents of the Mahonian word.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001840via SetPartition✓↗

MahonianWordDescents(partition)Descents 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 Count(Filter(
                                                Range(
                                                       1,
                                                       Length(Flatten(_x)) - 1,
                                                     ),
                                                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], i + 1) else 0,
                                                                                                  q,
                                                                                                ), Range(1, Length(_x)))), c),
                                                                     )), i),
                                              ))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for {{1}}

live ↗

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

Known value · FindStat St001840

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

live ↗

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

Known value · FindStat St001840

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

live ↗

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

Known value · FindStat St001840

About

Examples	5
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