Distributions of Inversions and Descents over Integer Compositions

TL;DR

This paper derives generating functions for counting integer compositions of n into k parts with specified inversions or descents, linking these distributions to permutation statistics via a known bijection.

Contribution

It introduces a novel connection between compositions' inversion and descent distributions and permutation statistics using a bijection.

Findings

Derived generating functions for compositions with fixed inversions or descents.

Established a relationship between composition distributions and permutation statistics.

Connected composition distributions to classical permutation statistics maj, inv, des.

Abstract

We derive a generating function for the number of integer compositions of $n$ into $k$ parts (i.e., $k$-compositions of $n$) with a given number of inversions, and obtain similar results for $k$-compositions of $n$ with a given number of descents. Our approach relies on a known bijection that associates each integer composition $\sigma$ with a pair $(\pi,\lambda)$, where $\pi$ is a permutation and $\lambda$ is an integer partition. We show that the distribution of inversions and the distribution of descents over $k$-compositions are related, respectively, to the distribution of (maj,inv) and to the distribution of (inv,des) over permutations of $\{1,2,\ldots,k\}$, where maj, inv, and des denote the classical permutation statistics major index, inversion number, and descent number, respectively.