Gamma-positivity for a refinement of median Genocchi numbers (Q2138572)

Summary: We study the generating function of descent numbers for the permutations with descent pairs of prescribed parities, the distribution of which turns out to be a refinement of median Genocchi numbers. We prove the \(\gamma\)-positivity for the polynomial and derive the generating function for the \(\gamma\)-vectors, expressed in the form of continued fraction. We also come up with an artificial statistic that gives a \(q\)-analogue of the \(\gamma\)-positivity for the permutations with descents only allowed from an odd value to an odd value.