GlobalAscents

Positions i < n with p(j) < p(k) whenever j <= i < k.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000234via Permutation✓↗

GlobalAscents(p)Positions i < n with p(j) < p(k) whenever j <= i < k.

Typed signatures

Type	Package	Applies
(list<integer> | 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 also a bare list of integers: this reading compares entries with each other rather than with their positions, so it stands on any sequence.
• Read through the pigeonhole principle: the largest of the first i entries is at most i.

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(Max(Map(
                                                         Function(_x[j], j),
                                                         Range(1, i),
                                                       )) <= i, i),
                                     ))

Examples

Basic #

the one-line word $312$

live ↗

In$\mathrm{GlobalAscents}({\operatorname{Permutation}}([3, 1, 2]))$Out$0$

the one-line word $312$, as a plain list

live ↗

In$\mathrm{GlobalAscents}(\bigl\lbrack3, 1, 2\bigr\rbrack)$Out$0$

the one-line word $2413$

live ↗

In$\mathrm{GlobalAscents}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$0$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000234

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000234

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000234

About

Examples	6
Crosswalk	FindStat