  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.
