OccurrencesOf14Dash2Dash3Or14Dash3Dash2

Adjacent ascents p(i) < p(i+1) with two later entries between them, in either order, counted in pairs.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000709via Permutation✓↗

OccurrencesOf14Dash2Dash3Or14Dash3Dash2(p)Adjacent ascents p(i) < p(i+1) with two later entries between them, in either order, counted in pairs.

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.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 3 else Sum(Map(
                                  Function(Apply(
                                                  (c: number) => c * (c - 1) / 2,
                                                  Count(Filter(
                                                                Range(
                                                                       i + 2,
                                                                       Length(_x),
                                                                     ),
                                                                Function(_x[i] <
                                                                         _x[j] && _x[j] <
                                                                             _x[i + 1], j),
                                                              )),
                                                ) if _x[i] < _x[i + 1] else 0, i),
                                  Range(1, Length(_x) - 2),
                                ))

Examples

Basic #

the one-line word $312$

live ↗

In$\mathrm{OccurrencesOf14Dash2Dash3Or14Dash3Dash2}({\operatorname{Permutation}}([3, 1, 2]))$Out$0$

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

live ↗

In$\mathrm{OccurrencesOf14Dash2Dash3Or14Dash3Dash2}(\bigl\lbrack3, 1, 2\bigr\rbrack)$Out$0$

the one-line word $2413$

live ↗

In$\mathrm{OccurrencesOf14Dash2Dash3Or14Dash3Dash2}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$0$

Properties #

FindStat's value for [1,2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2]), \text{OccurrencesOf14Dash2Dash3Or14Dash3Dash2})$Out$0$

Known value · FindStat St000709

FindStat's value for [2,1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{OccurrencesOf14Dash2Dash3Or14Dash3Dash2})$Out$0$

Known value · FindStat St000709

FindStat's value for [1,2,3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2, 3]), \text{OccurrencesOf14Dash2Dash3Or14Dash3Dash2})$Out$0$

Known value · FindStat St000709

About

Examples	6
Crosswalk	FindStat