Fa\`a di Bruno's formula and inversion of power series

Fa\`a di Bruno's formula gives an expression for the derivatives of the composition of two real-valued functions. In this paper we prove a multivariate and synthesized version of Fa\`a di Bruno's formula in higher dimensions, providing a combinatorial expression for the derivatives of chain compositions $F^{(1)} \circ \ldots \circ F^{(m)}$ of functions $F^{(l)} : \mathbb{R}^N \to \mathbb{R}^N$ in terms of sums over labelled trees. We give several applications of this formula, including a new involution formula for the inversion of multivariate power series. We use this framework to outline a combinatorial approach to studying the invertibility of polynomial mappings, giving a purely combinatorial restatement of the Jacobian conjecture. Our methods extend naturally to the non-commutative case, where we prove a free version of Fa\`a di Bruno's formula for multivariate power series in free indeterminates, and use this formula as a tool for obtaining a new inversion formula for free power series.