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We study Dirichlet series enumerating orbits of Cartesian products of mapswhose orbit distributions are modelled on the distributions of finite indexsubgroups of free abelian groups of finite rank. We interpret Euler factors ofsuch orbit Dirichlet series in terms of generating polynomials for statisticson 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 orbitDirichlet series under consideration. Secondly, we determine these (global)Dirichlet series' abscissae of convergence and establish some meromorphiccontinuation beyond these abscissae. As a corollary, we describe theasymptotics of the relevant orbit growth sequences. For Cartesian products ofmore than two maps we establish a natural boundary for meromorphiccontinuation. For products of two maps, we prove the existence of such anatural boundary subject to a combinatorial conjecture.

Orbit Dirichlet series and multiset permutations