Physical Sciences and Mathematics Commons^™

Higher Dimensional Lattice Chains And Delannoy Numbers, John S. Caughman, Charles L. Dunn, Nancy Ann Neudauer, Colin L. Starr

Faculty Publications

Fix nonnegative integers n₁ , . . ., n_d, and let L denote the lattice of points (a₁ , . . ., a_d) ∈ ℤ^d that satisfy 0 ≤ a_i ≤ n_i 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 n_i = n (for all i) in these expressions yields a new …