Journal
DISCRETE MATHEMATICS
Volume 308, Issue 12, Pages 2623-2628Publisher
ELSEVIER SCIENCE BV
DOI: 10.1016/j.disc.2007.05.017
Keywords
lattice; Delannoy numbers
Categories
Ask authors/readers for more resources
Fix nonnegative integers n(1),...,n(d) and let L denote the lattice of integer points (a(1),...,a(d)) epsilon Z(d) satisfying 0 <= a(i) <= n(i) 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 n(1) = n(2) 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. (c) 2007 Elsevier B.V. All rights reserved.
Authors
I am an author on this paper
Click your name to claim this paper and add it to your profile.
Reviews
Recommended
No Data Available