OuterPeaks

Peaks of the word 0 p(1) ... p(n) 0: the interior peaks, plus one for each end that is a peak.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000092via Permutation✓↗

OuterPeaks(p)Peaks of the word 0 p(1) ... p(n) 0: the interior peaks, plus one for each end that is a peak.

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

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000092

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000092

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000092

About

Examples	6
Crosswalk	FindStat