An Asymptotic Expansion for the Number of Permutations with a Certain Number of Inversions

Abstract

Let $b(n,k)$ denote the number of permutations of $\{1,\ldots,n\}$ with precisely $k$ inversions. We represent $b(n,k)$ as a real trigonometric integral and then use the method of Laplace to give a complete asymptotic expansion of the integral. Among the consequences, we have a complete asymptotic expansion for $b(n,k)/n!$ for a range of $k$ including the maximum of the $b(n,k)/n!$.