The number of permutations with k inversions

Abstract

Let $n\ge 1$, $0\le t\le \left(\genfrac{}{}{0ex}{}{n}{2}\right)$ be arbitrary integers. Define the numbers $I_n(t)$ as the number of permutations of $[n]$ with $t$ inversions. Let $n,d\ge 1$ and $0\le t\le (d-1)n$ be arbitrary integers. Define {\em the polynomial coefficients} $H(n,d,t)$ as the numbers of compositions of $t$ with at most $n$ parts, no one of which is greater than $d-1$. In our article we give explicit formulas for the numbers $I_n(t)$ and $H(n,d,t)$ using the theory of Gröbner bases and free resolutions.

arXiv comment20 pages, submitted to J. of Int. Sequences