95 v 3 1 9 O ct 2 00 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 all permutations of [n] := {1, . . . , n}. We say that 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 polynomial