  In the combinatorial theory of continued fractions, the Foata--Zeilberger
bijection and its variants have been extensively used to derive various
continued fractions enumerating several (sometimes infinitely many)
simultaneous statistics on permutations (combinatorial model for factorials)
and D-permutations (combinatorial model for Genocchi and median Genocchi
numbers). A Laguerre digraph is a digraph in which each vertex has in- and
out-degrees $0$ or $1$. In this paper, we interpret the Foata--Zeilberger
bijection in terms of Laguerre digraphs, which enables us to count cycles in
permutations. Using this interpretation, we obtain Jacobi-type continued
fractions for multivariate polynomials enumerating permutations, and also
Thron-type and Stieltjes-type continued fractions for multivariate polynomials
enumerating D-permutations, in both cases including the counting of cycles.
This enables us to prove some conjectured continued fractions due to Sokal and
Zeng (2022 Advances in Applied Mathematics) in the case of permutations, and
Randrianarivony and Zeng (1996 Electronic Journal of Combinatorics) and Deb and
Sokal (2022 arXiv) in the case of D-permutations.
