FlattenedAscentTopsAfterSmaller

Ascent tops of the flattened word with every smaller element earlier.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001641via SetPartition✓↗

FlattenedAscentTopsAfterSmaller(partition)Ascent tops of the flattened word with every smaller element earlier.

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.
• The binary logarithm of the number of set partitions with the same flattened word.

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(Flatten(_x)[i] <
                                                         Flatten(_x)[i + 1] && Count(Filter(
                                                                           Range(1, i),
                                                                           Function(Flatten(_x)[j] <
                                                                                    Flatten(_x)[i + 1], j),
                                                                         )) == Flatten(_x)[i + 1] -
                                                                              1, i),
                                              ))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for {{1}}

live ↗

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

Known value · FindStat St001641

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

live ↗

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

Known value · FindStat St001641

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

live ↗

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

Known value · FindStat St001641

About

Examples	5
Crosswalk	FindStat