  We discover that tautological intersection numbers on $\bar{\mathcal{M}}_{g,
n}$, the moduli space of stable genus $g$ curves with $n$ marked points, are
evaluations of Ehrhart polynomials of partial polytopal complexes. In order to
prove this, we realize the Virasoro constraints for tautological intersection
numbers as a recursion for integer-valued polynomials. Then we apply a theorem
of Breuer that classifies Ehrhart polynomials of partial polytopal complexes by
the nonnegativity of their $f^*$-vector. In dimensions 1 and 2, we show that
the polytopal complexes that arise are \emph{inside-out polytopes} i.e.
polytopes that are dissected by a hyperplane arrangement.
