AscentVariation

The sum of c(i+1) - c(i) over the positions where it is positive.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000091via IntegerComposition✓↗

AscentVariation(composition)The sum of c(i+1) - c(i) over the positions where it is positive.

Typed signatures

Type	Package	Applies
(composition) -> number	combinatorics	always

Domain: Composition statistics

Details

• Defined over IntegerComposition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a IntegerComposition 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.
• Archibald, Knopfmacher and Mansour, Variation statistics on compositions. FindStat titles it the descent variation; it sums the rises, which is equidistributed with the sum of the falls under reversal.

Definition

Epsil, as the implementation runs it.

on IntegerComposition, over _x

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

Examples

Basic #

the composition $[2,1,3,1,1]$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000091

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([2]), \text{AscentVariation})$Out$0$

Known value · FindStat St000091

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([3]), \text{AscentVariation})$Out$0$

Known value · FindStat St000091

About

Examples	4
Crosswalk	FindStat