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.