Gamma-positivity for a Refinement of Median Genocchi Numbers

2021/03/16 by Eu, Sen-Peng, Fu, Tung-Shan, Lai, Hsin-Hao +1 · 1 citation
#05A05 #05A15 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2103.09130

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 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.

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• Parity patterns meet Genocchi numbers, I: four labelings and three bijections

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