MULTIVARIABLE LAGRANGE INVERSION, GESSEL-VIENNOT CANCELLATION, AND THE MATRIX TREE THEOREM

A new form of multivariable Lagrange inversion is given, with determin ants occurring on both sides of the equality. These determinants are p rincipal miners, for complementary subsets of row and column indices, of two determinants that arise singly in the best known forms of multi variable Lagrange inversion. A combinatorial proof is given by conside ring functional digraphs, in which one of the principal miners is inte rpreted as a Matrix Tree determinant, and the other by a form of Gesse l-Viennot cancellation. (C) 1997 Academic Press.