Gamma-positivity for a Refinement of Median Genocchi Numbers

TL;DR

This paper explores the gamma-positivity of a polynomial related to median Genocchi numbers, providing new generating functions and a q-analogue for specific permutation statistics.

Contribution

It establishes gamma-positivity for a refined polynomial of median Genocchi numbers and derives its generating function in continued fraction form.

Findings

Proves gamma-positivity for the polynomial

Derives generating function for gamma-vectors

Introduces a q-analogue for certain permutation descents

Abstract

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.