ValleysIncludingBoundary

Positions lower than both neighbours, a larger entry standing off each end.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000099via Permutation✓↗

ValleysIncludingBoundary(p)Positions lower than both neighbours, a larger entry standing off each end.

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.
• Excluding the boundary is Valleys.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 1 else Count(Filter(
                                       Range(1, Length(_x)),
                                       Function((True if i == 1 else _x[i - 1] >
                                                                     _x[i]) &&
                                                (True if i == Length(_x) else _x[i] <
                                                                              _x[i + 1]), i),
                                     ))

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000099

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000099

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000099

About

Examples	5
Crosswalk	FindStat