AscentTopsWithSmallerValuesBefore

Ascents i whose top p(i+1) has every smaller value among p(1), ..., p(i).

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001640via Permutation✓↗

AscentTopsWithSmallerValuesBefore(p)Ascents i whose top p(i+1) has every smaller value among p(1), ..., 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.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

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

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001640

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001640

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001640

About

Examples	5
Crosswalk	FindStat