Multiple Lie derivatives and forests

We obtain a complete time expansion of the pull-back operator generated by a real analytic flow of real analytic automorphisms acting on analytic tensor sections of a manifold. Our expansion is given in terms of multiple Lie derivatives. Motivated by this expansion, we provide a rather simple and explicit estimate for higher order covariant derivatives of multiple Lie derivatives acting on smooth endomorphism sections of the tangent bundle of a manifold. We assume the covariant derivative to be torsion free. The estimate is given in terms of Dyck polynomials. The proof uses a new result on the combinatorics of rooted labeled ordered forests and Dyck polynomials.