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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.