Partial difference equations in m1⩾ m2⩾⋯⩾ mn⩾0 and their applications to combinatorics

Abstract

Various discrete functions encountered in Combinatorics are solutions of Partial Difference Equations in the subset of N n given by m 1⩾ m 2⩾⋯⩾ m n ⩾0. Given a partial difference equation, it is described how to pass from the standard “easy” solution of an equation in N n to a solution of the same equation subject to certain “Dirichlet” or “Neumann” boundary conditions in the domain m 1⩾ m 2⩾⋯⩾ m n ⩾0 and related domains. Applications include a rather quick derivation of MacMahon's generating function for plane partitions, a generalization and q-analog of the Ballot problem, and a joint analog of the Ballot problem and Simon Newcomb's problem.