Multivariable Lagrange Inversion, Gessel-Viennot Cancellation, and the Matrix Tree Theorem

A new form of multivariable Lagrange inversion is given, with determinants occurring on both sides of the equality. These determinants are principal minors, for complementary subsets of row and column indices, of two determinants that arise singly in the best known forms of multivariable Lagrange inversion. A combinatorial proof is given by considering functional digraphs, in which one of the principal minors is interpreted as a Matrix Tree determinant, and the other by a form of Gessel-Viennot cancellation.