Higher order log-concavity in Euler’s difference table

• Published in: Discrete mathematics 2011-10, Vol.311 (20), p.2128-2134
• Main Authors: Chen, William Y.C., Gu, Cindy C.Y., Ma, Kevin J., Wang, Larry X.W.
• Format: Article
• Language: English
• Subjects: 2-log-concavity; Algebra; Combinatorial analysis; Combinatorics; Combinatorics. Ordered structures; Euler’s difference table; Exact sciences and technology; Global analysis, analysis on manifolds; Log-concavity; Mathematical analysis; Mathematics; Permutations; Reverse ultra log-concavity; Sciences and techniques of general use; Tables (data); Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2011.06.006

For 0≤k≤n, let enk be the entries in Euler’s difference table and let dnk=enk/k!. Dumont and Randrianarivony showed enk equals the number of permutations on [n] whose fixed points are contained in {1,2,…,k}. Rakotondrajao found a combinatorial interpretation of the number dnk in terms of k-fixed-points-permutations of [n]. We show that for any n≥1, the sequence {dnk}0≤k≤n is essentially 2-log-concave and reverse ultra log-concave. ► In this paper, we study the higher order log-concavity of {dnk}0≤k≤n, where dnk=enk/n! and enk are the entries in Euler’s difference table. ► We show that the sequence {dnk}0≤k≤n is essentially 2-log-concave for any n≥1. ► We also show that the sequence {dnk}0≤k≤n is reverse ultra log-concave for any n≥1.