Twisted identities in Coxeter groups
Journal of Algebraic Combinatorics: An International Journal
Enumerative Combinatorics: Volume 1
Enumerative Combinatorics: Volume 1
The Eulerian distribution on involutions is indeed unimodal
Journal of Combinatorial Theory Series A
Actions on permutations and unimodality of descent polynomials
European Journal of Combinatorics
The descent statistic on involutions is not log-concave
European Journal of Combinatorics
European Journal of Combinatorics
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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 d, maj and the joint distribution (d, maj). We show that exc is log-concave 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 the light of these new results, we present further conjectures at the end of the paper.