In this article we describe cell decompositions of the moduli space of
Riemann surfaces and their relationship to a Hurwitz problem. The cells possess
natural linear structures and with respect to this they can be described as
rational convex polytopes which come equipped with natural integer points and a
volume form. We show how to effectively calculate the number of lattice points
and the volumes over all the cells and that these calculations contain deep
information about the moduli space.