Cycles of even-odd drop permutations and continued fractions of Genocchi numbers

TL;DR

This paper explores the combinatorial structures of Genocchi numbers through permutation models, confirming conjectures and introducing a new $(p,q)$-analogue of descent polynomials using continued fractions.

Contribution

It merges previous models and results to confirm Lazar-Wachs' conjecture and introduces a novel $(p,q)$-analogue of descent polynomials with gamma-positivity properties.

Findings

Confirmed Lazar-Wachs' conjecture on cycle distributions.

Introduced a $(p,q)$-analogue of descent polynomials.

Established gamma-positivity and factorization properties.

Abstract

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 $\gamma$-coefficients of our $(p,q)$-analogue of descent polynomials have the same factorization flavor as the $\gamma$-coeffcients of Br\"and\'en's $(p,q)$-Eulerian polynomials.