Permutation statistics on involutions

In this paper we look at polynomials arising from statistics on the classes of involutions, I_n, and involutions with no fixed points, J_n, in the symmetric group. Our results are motivated by Brenti's conjecture [F. Brenti, Private communication, 2004] which states that the Eulerian distribution of I_n is log-concave. Symmetry of the generating functions is shown for the statistics d, maj and the joint distribution (d, maj). We show that exc is log-concave on I_n, inv is log-concave on J_n and d is partially unimodal on both I_n and J_n. We also give recurrences and explicit forms for the generating functions of the inversions statistic on involutions in Coxeter groups of types B_n and D_n. Symmetry and unimodality of inv is shown on the subclass of signed permutations in D_n with no fixed points. In the light of these new results, we present further conjectures at the end of the paper.