Higher Order Log-Concavity in Euler's Difference Table

TL;DR

This paper investigates the higher order log-concavity properties of a permutation-related array derived from Euler's difference table, revealing new concavity characteristics in combinatorial sequences.

Contribution

It establishes that the array formed by scaled entries of Euler's difference table exhibits 2-log-concavity and reverse ultra log-concavity for all n, a novel insight into their combinatorial structure.

Findings

The sequence is 2-log-concave for all n.

The sequence is reverse ultra log-concave for all n.

Provides new combinatorial properties of Euler's difference table entries.

Abstract

Let $e_{n}^k$ be the entries in the classical Euler's difference table. We consider the array $d_{n}^{k}=e_n^k/k!$ for $0\leq k \leq n$, where $d_n^k$ can be interpreted as the number of k-fixed-points-permutations of [n]. We show that the sequence $\{d_n^k\}_{0\leq k\leq n}$ is 2-log-concave and reverse ultra log-concave for any given n.