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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 theGenocchi and median Genocchi numbers. We find Stieltjes-type and Thron-typecontinued fractions for some multivariate polynomials that enumerateD-permutations with respect to a very large (sometimes infinite) number ofsimultaneous statistics that measure cycle status, record status, crossings andnestings.

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