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We show that the Poincare polynomial associated with the orbifold celldecomposition of the moduli space of smooth algebraic curves with distinctmarked points satisfies a topological recursion formula of the Eynard-Orantintype. The recursion uniquely determines the Poincare polynomials from theinitial data. Our key discovery is that the Poincare polynomial is the Laplacetransform of the number of Grothendieck's dessins d'enfants.

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