Successions

Permutation: positions i with p(i+1) = p(i) + 1. SetPartition: elements i with i + 1 in the same block.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000164via PerfectMatching✓↗St000441via Permutation✓↗St000502via SetPartition✓↗

Successions(p)Permutation: positions i with p(i+1) = p(i) + 1. SetPartition: elements i with i + 1 in the same block.

Typed signatures

Type	Package	Applies
(list<integer> | permutation | set_partition) -> number	combinatorics	always

Domain: Permutation statistics

Details

• Defined over Permutation as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a Permutation, and also a bare list of integers: this reading compares entries with each other rather than with their positions, so it stands on any sequence.
• 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.
• Also called small ascents or 1-rises.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 2 else Count(Filter(
                                       Range(1, Length(_x) - 1),
                                       Function(_x[i + 1] == _x[i] + 1, i),
                                     ))

on SetPartition, over _x

0 if Length(_x) < 1 else 0 if Length(Flatten(Map(Function(Most(b), b), _x))) < 1 else Count(Filter(
                                                                                                    Range(1, Length(Flatten(Map(Function(Most(b), b), _x))), 1),
                                                                                                    Function(Flatten(Map(Function(Rest(b), b), _x))[i] -
                                                                                                             Flatten(Map(Function(Most(b), b), _x))[i] ==
                                                                                                             1, i),
                                                                                                  ))

Examples

Basic #

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

live ↗

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

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

live ↗

In$\mathrm{Successions}({\operatorname{SetPartition}}([[1], [2, 4], [3]]))$Out$0$

the one-line word $312$

live ↗

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

the one-line word $312$, as a plain list

live ↗

In$\mathrm{Successions}(\bigl\lbrack3, 1, 2\bigr\rbrack)$Out$1$

the one-line word $2413$

live ↗

In$\mathrm{Successions}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$0$

Properties #

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

live ↗

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

Known value · FindStat St000502

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

live ↗

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

Known value · FindStat St000502

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

live ↗

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

Known value · FindStat St000502

FindStat's value for [1]

live ↗

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

Known value · FindStat St000441

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000441

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000441

About

Examples	11
Crosswalk	FindStat ×3