Valleys

The number of valleys of $p$ : interior positions $i$ with $p_{i-1} > p_i < p_{i+1}$ .

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000053via DyckPath✓St001169via DyckPath✓St000353via Permutation✓

Valleys(p)the valley count of a one-line permutation

p

Typed signatures

Type	Package	Applies
`(dyck_path \| list \| permutation) -> integer \| number`	`combinatorics`	always

Domain: Permutation statistics

Details

Only interior positions count ( $1 < i < n$ ), so $\mathrm{Valleys}(p) = 0$ whenever $n \leq 2$
The local minima, complementary in shape to the local maxima counted by Peaks
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.
A separate definition exists for DyckPath (Occurrences of a down step immediately followed by an up step.) but is not the one declared: one head, one owner.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 3 else Count(Filter(
                                       Range(2, Length(_x) - 1),
                                       Function(_x[i - 1] > _x[i] && _x[i] <
                                                                     _x[i + 1], i),
                                     ))

on DyckPath, over _x

0 if Length(_x) < 2 else Count(Filter(
                                       Range(1, Length(_x) - 1),
                                       Function(_x[i] == 0 && _x[i + 1] == 1, i),
                                     ))