The Lagrange Inversion Formula and Divisibility Properties

Wilf stated that the Lagrange inversion formula (LIF) is a remarkable tool for solving certain kinds of functional equations, and at its best it can give explicit formulas where other approaches run into stone walls. Here we present the LIF combinatorially in the form of lattice paths, and apply it to the divisibility property of the coefficients of a formal power series expansion. For the LIF, the coefficients are in a commutative ring with identity. As for divisibility, we require the coefficients to be in a principal ideal domain.