The degree of asymmetry of sequences

TL;DR

This paper introduces a new combinatorial statistic called the degree of asymmetry for various integer sequences and objects, providing generating functions and analyzing their limiting distributions.

Contribution

It defines the degree of asymmetry for multiple combinatorial objects and derives their generating functions and asymptotic distributions.

Findings

Generated enumerations for compositions, words, matchings, binary trees, permutations

Derived limiting distributions of the asymmetry degree

Provided new insights into symmetry properties of combinatorial objects

Abstract

We explore the notion of degree of asymmetry for integer sequences and related combinatorial objects. The degree of asymmetry is a new combinatorial statistic that measures how far an object is from being symmetric. We define this notion for compositions, words, matchings, binary trees and permutations, we find generating functions enumerating these objects with respect to their degree of asymmetry, and we describe the limiting distribution of this statistic in each case.