Permutation statistics on involutions

In this paper we look at polynomials arising from statistics on the classes of involutions, In, and involutions with no fixed points, Jn, in the symmetric group. Our results are motivated by F. Brenti’s conjecture [3] which states that the Eulerian distribution of In is logconcave. Symmetry of the generating functions is shown for the statistics d, maj and the joint distribution (d, maj). We show that exc is logconcave on In, inv is log-concave on Jn and d is partially unimodal on both In and Jn. We also give recurrences and explicit forms for the generating functions of the inversions statistic on involutions in Coxeter groups of types Bn and Dn. Symmetry and unimodality of inv is shown on the subclass of signed permutations in Dn with no fixed points. In light of these new results, we present further conjectures at the end of the paper.