CyclePeaks

Values i with p^-1(i) < i > p(i).

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000994via Permutation✓↗

CyclePeaks(p)Values i with p^-1(i) < i > p(i).

Typed signatures

Type	Package	Applies
(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 nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer.
• Every cycle has as many peaks as valleys, so this is the cycle valleys too.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 1 else Count(Filter(
                                       Range(1, Length(_x)),
                                       Function(
                                                 Sum(Map(
                                                          Function(
                                                                    j if _x[j] ==
                                                                         i else 0,
                                                                    j,
                                                                  ),
                                                          Range(1, Length(_x)),
                                                        )) < i && _x[i] < i,
                                                 i,
                                               ),
                                     ))

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000994

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000994

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000994

About

Examples	5
Crosswalk	FindStat