The Eulerian distribution on centrosymmetric involutions

We present an extensive study of the Eulerian distribution on the set of centrosymmetric involutions, namely, involutions in Sn satisfying the property σ(i) + σ(n+ 1− i) = n+ 1 for every i = 1 . . . n. We find some combinatorial properties for the generating polynomial of such distribution, together with an explicit formula for its coefficients. Afterwards, we carry out an analogous study for the subset of centrosymmetric involutions without fixed points.