Partial difference equations in m1?m2???mn?0 and their applications to combinatorics |
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Authors: | Doron Zeilberger |
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Institution: | Department of Mathematics, University of Illinois, Urbana-Champaign, IL 61801, USA |
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Abstract: | Various discrete functions encountered in Combinatorics are solutions of Partial Difference Equations in the subset of Nn given by m1?m2???mn?0. Given a partial difference equation, it is described how to pass from the standard “easy” solution of an equation in Nn to a solution of the same equation subject to certain “Dirichlet” or “Neumann” boundary conditions in the domain m1?m2???mn?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. |
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