  We study Dirichlet series enumerating orbits of Cartesian products of maps
whose orbit distributions are modelled on the distributions of finite index
subgroups of free abelian groups of finite rank. We interpret Euler factors of
such orbit Dirichlet series in terms of generating polynomials for statistics
on multiset permutations, viz. descent and major index, generalizing Carlitz's
$q$-Eulerian polynomials.
  We give two main applications of this combinatorial interpretation. Firstly,
we establish local functional equations for the Euler factors of the orbit
Dirichlet series under consideration. Secondly, we determine these (global)
Dirichlet series' abscissae of convergence and establish some meromorphic
continuation beyond these abscissae. As a corollary, we describe the
asymptotics of the relevant orbit growth sequences. For Cartesian products of
more than two maps we establish a natural boundary for meromorphic
continuation. For products of two maps, we prove the existence of such a
natural boundary subject to a combinatorial conjecture.
