  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.
