The Eulerian distribution on involutions is indeed unimodal

Let In,k (respectively Jn,k) be the number of involutions (respectively fixed-point free involutions) of {1,...,n} with k descents. Motivated by Brenti's conjecture which states that the sequence In,0, In,1,..., In,n-1 is log-concave, we prove that the two sequences In,k and J2n,k are unimodal in k, for all n. Furthermore, we conjecture that there are nonnegative integers an,k such that Σk=0n-1 In,ktk= Σk=0⌊(n-1)/2⌋ an,ktk(1+t)n-2k-1.This statement is stronger than the unimodality of In,k but is also interesting in its own right.