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

Abstract

Recently Lazar and Wachs proved two new permutation models, called D-permutations and E-permutations, for Genocchi and median Genocchi numbers. In a follow-up, Eu et al. studied the even-odd descent permutations, which are in bijection with E-permutations. We generalize Eu et al.'s descent polynomials with eight statistics and obtain an explicit J-fraction formula for their ordinary generaing function. The J-fraction permits us to confirm two conjectures of Lazar-Wachs about cycles of D and E permutations and obtain a $(p,q)$-analogue of Eu et al.'s gamma-formula. Moreover, the $(p,q)$ gamma-coefficients have the same factorization flavor as the gamma-coefficients of Brändén's $(p,q)$-Eulerian polynomials.