2021/08/06 by Pan, Qiongqiong, Zeng, Jiang · 1 citation
#05A19 #30B70 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2108.03200
Recently, Lazar and Wachs (arXiv:1910.07651) showed that the (median) Genocchi numbers play a fundamental role in the study of the homogenized Linial arrangement and obtained two new permutation models (called D-permutations and E-permutations) for (median) Genocchi numbers. They further conjecture that the distributions of cycle numbers over the two models are equal. In a follow-up, Eu et al. (arXiv:2103.09130) further proved the gamma-positivity of the descent polynomials of even-odd descent permutations, which are in bijection with E-permutations by Foata's fundamental transformation. This paper merges the above two papers by considering a general moment sequence which encompasses the number of cycles and number of drops of E-permutations. Using the combinatorial theory of continued fraction, the moment connection enables us to confirm Lazar-Wachs' conjecture and obtain a natural (p,q)-analogue of Eu et al's descent polynomials. Furthermore, we show that the γ-coefficients of our (p,q)-analogue of descent polynomials have the same factorization flavor as the γ-coeffcients of Brändén's (p,q)-Eulerian polynomials.