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Counting Lattice Chains and Delannoy Paths in Higher Dimensions
Discrete Mathematics (2011)
  • John S Caughman, IV, Portland State University
  • Charles L. Dunn, LInfield College
  • Nancy Ann Neudauer, Pacific University
  • Colin L. Starr, Willamette University
Lattice chains and Delannoy paths represent two different ways to progress through a lattice. We use elementary combinatorial arguments to derive new expressions for the number of chains and the number of Delannoy paths in a lattice of arbitrary finite dimension. Specifically, fix nonnegative integers , and let denote the lattice of points that satisfy for . We prove that the number of chains in is given by where . We also show that the number of Delannoy paths in equals Setting (for all ) in these expressions yields a new proof of a recent result of Duchi and Sulanke relating the total number of chains to the central Delannoy numbers. We also give a novel derivation of the generating functions for these numbers in arbitrary dimension.
Publication Date
August, 2011
Citation Information
John S Caughman, Charles L. Dunn, Nancy Ann Neudauer and Colin L. Starr. "Counting Lattice Chains and Delannoy Paths in Higher Dimensions" Discrete Mathematics Vol. 311 Iss. 16 (2011)
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