AscentTopSum

The sum of the ascent tops: p(i+1) over the ascents i.

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AscentTopSum(p)The sum of the ascent tops: p(i+1) over the ascents i.

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.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

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

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

In$\mathrm{AscentTopSum}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$7$

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000471

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000471

FindStat's value for [1,2,3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2, 3]), \text{AscentTopSum})$Out$5$

Known value · FindStat St000471

About

Examples	6
Crosswalk	FindStat