CyclicValleys

Positions lower than both neighbours, reading the word cyclically.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000243via Permutation✓↗

CyclicValleys(p)Positions lower than both neighbours, reading the word cyclically.

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.
• As many cyclic peaks as cyclic valleys, so this is both.

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,
                                                                           1,
                                                                         ),
                                                                    Function(w[i -
                                                                               1] >
                                                                             w[i] &&
                                                                             w[i] <
                                                                             w[i +
                                                                               1], i),
                                                                  )),
                                Join([_x[Length(_x)]], _x, [_x[1]]),
                              )

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000243

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000243

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

live ↗

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

Known value · FindStat St000243

About

Examples	6
Crosswalk	FindStat