Gamma-Positivity for a Refinement of Median Genocchi Numbers

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 γ-positivity for the polynomial and derive the generating function for the γ-vectors, expressed in the form of continued fraction. We also come up with an artificial statistic that gives a q-analogue of the γ-positivity for the permutations with descents only allowed from an odd value to an odd value.