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Stein's method and the modular behavior of Eulerian numbers

Jason Fulman, Adrian Röllin

Abstract

The Eulerian number A(n,k) counts permutations of n symbols with exactly k descents. Motivated by questions 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 an explicit error bound for this convergence using Stein's method for translated Poisson approximation.

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 questions 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 an explicit error bound for this convergence using Stein's method for translated Poisson approximation.
Paper Structure (3 sections, 4 theorems, 29 equations)

This paper contains 3 sections, 4 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Acknowledgements

Key Result

Theorem 2.1

Suppose that $n \geq 6$. Let $b \ge 2$ and $k \in \{0,1,\dots,b-1\}$. Then

Theorems & Definitions (8)

  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Theorem 2.4: See Rol
  • Theorem 2.5
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}