Asymptotics of the Eulerian numbers revisited: A large deviation analysis

Abstract

Using the Saddle point method and multiseries expansions, we obtain from the generating function of the Eulerian numbers $A_{n,k}$ and Cauchy’s integral formula, asymptotic results in non-central region. In the region $k=n–n^{\alpha }$, $1>\alpha >1/2$, we analyze the dependence of $A_{n,k}$ on $\alpha$. This paper fits within the framework of Analytic Combinatorics.