The Eulerian Distribution on Involutions is Indeed Unimodal

Let I_{n,k} (resp. J_{n,k}) be the number of involutions (resp. fixed-point free involutions) of {1,…,n} with k descents. Motivated by Brenti’s conjecture which states that the sequence I_{n,0}, I_{n,1},…, I_{n,n-1} is log-concave, we prove that the two sequences I_{n,k} and J_{2n,k} are unimodal in k, for all n. Furthermore, we conjecture that there are nonnegative integers a_{n,k} such that $$\sum _{k=0}^{n-1}{I}_{n,k}{t}^k=\sum _{k=0}^{\lfloor (n-1)\mathrm{/}2\rfloor }{a}_{n,k}{t}^k(1+t{)}^{n-2k-1}\mathrm{.}$$ This statement is stronger than the unimodality of I_{n,k} but is also interesting in its own right.