Distributions of Inversions and Descents over Integer Compositions

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 $\{1,2,\dots ,k\}$, where maj, inv, and des denote the classical permutation statistics major index, inversion number, and descent number, respectively.