Log-concavity of the Excedance Enumerators in positive elements of Type A and Type B Coxeter Groups

Abstract

The classical Eulerian Numbers $A_{n,k}$ are known to be log-concave. Let $P_{n,k}$ and $Q_{n,k}$ be the number of even and odd permutations with $k$ excedances. In this paper, we show that $P_{n,k}$ and $Q_{n,k}$ are log-concave. For this, we introduce the notion of strong synchronisation and ratio-alternating which are motivated by the notion of synchronisation and ratio-dominance, introduced by Gross, Mansour, Tucker and Wang in 2014. We show similar results for Type B Coxeter Groups. We finish with some conjectures to emphasize the following: though strong synchronisation is stronger than log-concavity, many pairs of interesting combinatorial families of sequences seem to satisfy this property.

arXiv comment18 Pages, some typos of the last version corrected and more details added in the proof of Theorem 5