Document Type
Post-Print
Publication Date
2005
Subjects
Combinatorial probabilities, Lattice theory, Mathematical analysis, Lattice paths
Abstract
Fix nonnegative integers n1,…,nd and let L denote the lattice of integer points (a1,…,ad)∈Zd satisfying 0⩽ai⩽ni for 1⩽i⩽d. Let L be partially ordered by the usual dominance ordering. In this paper we offer combinatorial derivations of a number of results concerning chains in L. In particular, the results obtained are established without recourse to generating functions or recurrence relations. We begin with an elementary derivation of the number of chains in L of a given size, from which one can deduce the classical expression for the total number of chains in L. Then we derive a second, alternative, expression for the total number of chains in L when d=2. Setting n1=n2 in this expression yields a new proof of a result of Stanley [Enumerative Combinatorics, vol. 2, Cambridge University Press, Cambridge, 1999] relating the total number of chains to the central Delannoy numbers. We also conjecture a generalization of Stanley's result to higher dimensions.
DOI
10.1016/j.disc.2007.05.017
Persistent Identifier
http://archives.pdx.edu/ds/psu/17798
Citation Details
Caughman, John S. IV; Haithcock, Clifford R.; and Veerman, J. J. P., "A Note on Lattice Chains and Delannoy Numbers" (2005). Mathematics and Statistics Faculty Publications and Presentations. 145.
http://archives.pdx.edu/ds/psu/17798
Description
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 and can be found online at: http://dx.doi.org/10.1016/j.disc.2007.05.017