5 The Eulerian Distribution on Involutions is Indeed Unimodal

A sequence a0, a1, . . . , an of real numbers is said to be unimodal if for some 0 ≤ j ≤ n we have a0 ≤ a1 ≤ · · · ≤ aj ≥ aj+1 ≥ · · · ≥ an, and is said to be log-concave if a 2 i ≥ ai−1ai+1 for all 1 ≤ i ≤ n − 1. Clearly a log-concave sequence of positive terms is unimodal. The reader is referred to Stanley’s survey [10] for the surprisingly rich variety of methods to show that a sequence is log-concave or unimodal. As noticed by Brenti [2], even though log-concave and unimodality have one-line definitions, to prove the unimodality or logconcavity of a sequence can sometimes be a very difficult task requiring the use of intricate combinatorial constructions or of refined mathematical tools. Let Sn be the set of permutations of [n] := {1, . . . , n}. A permutation π = a1a2 · · ·an ∈ Sn has a descent at i (1 ≤ i ≤ n− 1) if ai > ai+1. The number of descents of π is called its descent number and is denoted by d(π). A statistic on Sn is said to be Eulerian, if it is equidistributed with the descent number statistic. Recall that the generating function of descent numbers on Sn is the Eulerian polynomial