AugmentedDoubleAscents

Positions i with w(i) < w(i+1) < w(i+2) in the word w = 0 p(1) ... p(n).

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001948via Permutation✓↗

AugmentedDoubleAscents(p)Positions i with w(i) < w(i+1) < w(i+2) in the word w = 0 p(1) ... p(n).

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) < 2 else Apply((w: list<integer>) => Count(Filter(
                                                                   Range(
                                                                          1,
                                                                          Length(w) -
                                                                          2,
                                                                          1,
                                                                        ),
                                                                   Function(
                                                                             w[i] <
                                                                             w[i +
                                                                               1] &&
                                                                             w[i +
                                                                               1] <
                                                                             w[i +
                                                                               2],
                                                                             i,
                                                                           ),
                                                                 )), Join(
                                                                          [0],
                                                                          _x,
                                                                        ))

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001948

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001948

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

live ↗

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

Known value · FindStat St001948

About

Examples	6
Crosswalk	FindStat