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doi:10.1016/j.ejc.2005.07.014 
Copyright © 2005 Elsevier Ltd All rights reserved.
Permutation statistics on involutions
W.M.B. Dukesa, 
Received 30 March 2005;
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Abstract
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 Brenti’s conjecture [F. Brenti, Private communication, 2004] which states that the Eulerian distribution of In is log-concave. Symmetry of the generating functions is shown for the statistics and the joint distribution
. We show that
is log-concave on In,
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
is shown on the subclass of signed permutations in Dn with no fixed points. In the light of these new results, we present further conjectures at the end of the paper.
