Permutation statistics on involutions

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