Stein's method and the modular behavior of Eulerian numbers

Abstract

The Eulerian number A(n,k) counts permutations of n symbols with exactly k descents. Motivated by problems in cryptography, several authors have studied the proportion of permutations whose number of descents lies in a fixed congruence class mod b, and its convergence to 1/b. We give two proofs of explicit error bounds for this convergence, one using Stein's method for translated Poisson approximation and one using Fourier analysis. The error bound using Fourier analysis yields exponentially decaying error bounds for fixed b, which generalises the already known case b=2; however, it makes use of a special representation due to Tanny (1973). In contrast, Stein's method only yields polynomially decaying error bounds, but we hope it has potential for generalisation beyond the present setting.