Topological recursion for the Poincaré polynomial of the combinatorial moduli space of curves

We show that the Poincaré polynomial associated with the orbifold cell decomposition of the moduli space of smooth algebraic curves with distinct marked points satisfies a topological recursion formula of the Eynard-Orantin type. The recursion uniquely determines the Poincaré polynomials from the initial data. Our key discovery is that the Poincaré polynomial is the Laplace transform of the number of Grothendieck's dessins d'enfants.