OccurrencesOfInitial12Dash3

Triples 1, 2, k ordered as 123 (|12-3).

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001084via Permutation✓↗

OccurrencesOfInitial12Dash3(p)Triples 1, 2, k ordered as 123 (|12-3).

Typed signatures

Type	Package	Applies
(list<integer> | permutation) -> 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.
• FindStat titles this |1-23, but its definition matches p(1) and p(2) as adjacent.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 3 else Count(Filter(Range(3, Length(_x)), Function(_x[1] <
                                                                     _x[2] && _x[2] <
                                                                         _x[k], k)))

Examples

Basic #

the one-line word $312$

live ↗

In$\mathrm{OccurrencesOfInitial12Dash3}({\operatorname{Permutation}}([3, 1, 2]))$Out$0$

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

live ↗

In$\mathrm{OccurrencesOfInitial12Dash3}(\bigl\lbrack3, 1, 2\bigr\rbrack)$Out$0$

the one-line word $2413$

live ↗

In$\mathrm{OccurrencesOfInitial12Dash3}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$0$

Properties #

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1]), \text{OccurrencesOfInitial12Dash3})$Out$0$

Known value · FindStat St001084

FindStat's value for [1,2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2]), \text{OccurrencesOfInitial12Dash3})$Out$0$

Known value · FindStat St001084

FindStat's value for [2,1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{OccurrencesOfInitial12Dash3})$Out$0$

Known value · FindStat St001084

About

Examples	6
Crosswalk	FindStat