Gamma-Positivity for a Refinement of Median Genocchi Numbers

  • Sen-Peng Eu
  • Tung-Shan Fu
  • Hsin-Hao Lai
  • Yuan-Hsun Lo


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.

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