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

TL;DR

This paper proves the log-concavity of excedance enumerators in positive elements of Type A and Type B Coxeter groups, introducing new concepts like strong synchronisation and ratio-alternating.

Contribution

It introduces the notions of strong synchronisation and ratio-alternating, proving log-concavity for excedance counts in Coxeter groups, extending classical results.

Findings

Log-concavity of P_{n,k} and Q_{n,k} in Type A Coxeter groups

Log-concavity of similar sequences in Type B Coxeter groups

Introduction of strong synchronisation and ratio-alternating concepts

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.