We discover that tautological intersection numbers on \(\overline{\mathcal{M}}_{g, n}\), 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 inside-out polytopes i.e. polytopes that are dissected by a hyperplane arrangement.

Mathematics Subject Classifications: 14H10, 52B20

Keywords: Moduli of curves, Ehrhart polynomials