Classical continued fractions for some multivariate polynomials generalizing the Genocchi and median Genocchi numbers

A D-permutation is a permutation of $[2n]$ satisfying $2k-1\le \sigma (2k-1)$ and $2k\ge \sigma (2k)$ for all $k$; they provide a combinatorial model for the Genocchi and median Genocchi numbers. We find Stieltjes-type and Thron-type continued fractions for some multivariate polynomials that enumerate D-permutations with respect to a very large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings.