Differential algebra of polytopes and inversion formulas

We use the differential algebra of polytopes to explain the known remarkable relation of the combinatorics of the associahedra and permutohedra with the universal compositional and multiplicative inversion formulas for the formal power series. This approach allows to single out the associahedra and permutohedra among all graph-associahedra and emphasizes the significance of the differential equations for special sequences of simple polytopes derived earlier by one of the authors. We discuss also the link with the geometry of Deligne-Mumford moduli spaces ${\bar{M}}_{0,n}$ and the interpretation of the combinatorics of cyclohedra in relation with the classical Faà di Bruno’s formula.