Discrete Mathematics and Combinatorics
Fix nonnegative integers n1 , . . ., nd, and let L denote the lattice of points (a1 , . . ., ad) ∈ ℤd that satisfy 0 ≤ ai ≤ ni for 1 ≤ i ≤ d. Let L be partially ordered by the usual dominance ordering. In this paper we use elementary combinatorial arguments to derive new expressions for the number of chains and the number of Delannoy paths in L. Setting ni = n (for all i) in these expressions yields a new proof of a recent result of Duichi 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.
This is the author’s version of a work that was accepted for publication in Discrete Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Discrete Mathematics as "Counting Lattice Chains and Delannoy Paths in Higher Dimensions," Volume 311, Issue 16, 2011, DOI: 10.1016/j.disc.2011.04.024
John S. Caughman, Charles L. Dunn, Nancy Ann Neudauer, & Colin L. Starr
Counting Lattice Chains and Delannoy Paths in Higher Dimensions.
Discrete Mathematics, 2011, volume 311, issue 16, pages 1803-1812
Caughman, John S.; Dunn, Charles L.; Neudauer, Nancy Ann; and Starr, Colin L., "Higher Dimensional Lattice Chains and Delannoy Numbers" (2011). Faculty Publications. Accepted Version. Submission 2.