AlternatingRuns

The interior peaks plus the interior valleys.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000483via Permutation✓↗

AlternatingRuns(p)The interior peaks plus the interior valleys.

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's title: the times the word switches from increasing to decreasing or back.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 3 else Apply((w: list<integer>) => Count(Filter(
                                                                   Range(
                                                                          2,
                                                                          Length(w) -
                                                                          1,
                                                                        ),
                                                                   Function(
                                                                             w[i -
                                                                               1] <
                                                                             w[i] &&
                                                                             w[i] >
                                                                             w[i +
                                                                               1],
                                                                             i,
                                                                           ),
                                                                 )), _x) + Apply(
                                                                                 (w: list<integer>) => Count(Filter(
                                                                                                                     Range(2, Length(w) - 1),
                                                                                                                     Function(w[i - 1] <
                                                                                                                              w[i] &&
                                                                                                                              w[i] >
                                                                                                                              w[i + 1], i),
                                                                                                                   )),
                                                                                 Map(
                                                                                      Function(Length(_x) + 1 -
                                                                                               _x[i], i),
                                                                                      Range(1, Length(_x)),
                                                                                    ),
                                                                               )

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000483

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000483

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000483

About

Examples	6
Crosswalk	FindStat