Ascents

The number of ascents of $p$ : positions $i$ with $p_i < p_{i+1}$ .

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaPermutationPermutationMathWorldPermutationAscentPermutationWikidataQ161519Encyclopedia of MathematicsPermutationnLabpermutationBritannicatopic/permutationFindStatSt000292via BinaryWord✓St000761via IntegerComposition✓St001935via ParkingFunction✓St000245via Permutation✓

Ascents(p)the ascent count of a one-line permutation

p

Typed signatures

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

Domain: Permutation statistics

Details

Complementary to Descents: $\mathrm{Ascents} + \mathrm{Descents} = n - 1$
The identity is all-ascent, the reversal all-descent
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.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

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

on IntegerComposition, over _x

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

on BinaryWord, over _x

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