Stein's method and the modular behavior of Eulerian numbers

TL;DR

This paper investigates the distribution of Eulerian numbers modulo b, providing explicit error bounds for their convergence to uniformity, using Stein's method and Fourier analysis.

Contribution

It offers two proofs of error bounds for the distribution of Eulerian numbers mod b, with potential for broader application.

Findings

Fourier analysis yields exponentially decaying error bounds for fixed b.

Stein's method provides polynomially decaying error bounds, with potential for generalization.

Results extend understanding of permutation descent distributions in cryptography.

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.