Distribution of descents in matchings

TL;DR

This paper investigates the distribution of descents in matchings, revealing symmetry properties and establishing asymptotic normality, thus advancing understanding of permutation statistics in combinatorics.

Contribution

It provides a bijective proof of symmetry and uses generating functions to prove asymptotic normality for descents in matchings.

Findings

Symmetry of descents and major indices in matchings

Asymptotic normality of the number of descents in matchings

Generating function approach to permutation statistics

Abstract

The distribution of descents in a fixed conjugacy class of $S_n$ is studied, and it is shown that its moments have an interesting property. A particular conjugacy class that is of interest is the class of matchings (also known as fixed point free involutions). This paper provides a bijective proof of the symmetry of the descents and major indices of matchings and uses a generating function approach to prove an asymptotic normality theorem for the number of descents in matchings.