Geodesic
Gamma-positivity for a refinement of median Genocchi numbers
Sen-Peng Eu1 ; Tung-Shan Fu2 ; Hsin-Hao Lai3 ; Yuan-Hsun Lo2
1National Taiwan Normal University; Chinese Air Force Academy
2National Pingtung University
3National Kaohsiung Normal University
The electronic journal of combinatorics, Tome 29 (2022) no. 2
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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.
DOI : 10.37236/10312
Classification : 05A05, 05A15
Mots-clés : generating function of descent numbers, Dumont permutations

Sen-Peng Eu  1   ; Tung-Shan Fu  2   ; Hsin-Hao Lai  3   ; Yuan-Hsun Lo  2

1 National Taiwan Normal University; Chinese Air Force Academy
2 National Pingtung University
3 National Kaohsiung Normal University 
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     author = {Sen-Peng Eu and Tung-Shan Fu and Hsin-Hao Lai and Yuan-Hsun Lo},
     title = {Gamma-positivity for a refinement of median {Genocchi} numbers},
     journal = {The electronic journal of combinatorics},
     year = {2022},
     volume = {29},
     number = {2},
     doi = {10.37236/10312},
     zbl = {1487.05011},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/10312/}
}
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Sen-Peng Eu; Tung-Shan Fu; Hsin-Hao Lai; Yuan-Hsun Lo. Gamma-positivity for a refinement of median Genocchi numbers. The electronic journal of combinatorics, Tome 29 (2022) no. 2. doi: 10.37236/10312

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