The Eulerian distribution on involutions is indeed unimodal

Let In,kIn,k (respectively Jn,kJn,k) be the number of involutions (respectively fixed-point free involutions) of {1,…,n}{1,…,n} with k   descents. Motivated by Brenti's conjecture which states that the sequence In,0,In,1,…,In,n−1In,0,In,1,…,In,n−1 is log-concave, we prove that the two sequences In,kIn,k and J2n,kJ2n,k are unimodal in k, for all n  . Furthermore, we conjecture that there are nonnegative integers an,kan,k such that∑k=0n−1In,ktk=∑k=0⌊(n−1)/2⌋an,ktk(1+t)n−2k−1. This statement is stronger than the unimodality of In,kIn,k but is also interesting in its own right.