AsymmetryDegree

Pairs of mirror positions holding different parts.

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AsymmetryDegree(composition)Pairs of mirror positions holding different parts.

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.

Definition

Epsil, as the implementation runs it.

on IntegerComposition, over _x

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

Examples

Basic #

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001673

FindStat's value for [2]

live ↗

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

Known value · FindStat St001673

FindStat's value for [3]

live ↗

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

Known value · FindStat St001673

About

Examples	4
Crosswalk	FindStat