FlattenedMajorIndex

The major index of the permutation listing the blocks in order, each ascending.

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FlattenedMajorIndex(partition)The major index of the permutation listing the blocks in order, each ascending.

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 Flatten(_x)[i] >
                                                         Flatten(_x)[i + 1] else 0, i),
                                           Range(1, Length(Flatten(_x)) - 1),
                                         ))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St000748

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

live ↗

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

Known value · FindStat St000748

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

live ↗

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

Known value · FindStat St000748

About

Examples	5
Crosswalk	FindStat