期刊
DISCRETE MATHEMATICS
卷 308, 期 12, 页码 2623-2628出版社
ELSEVIER SCIENCE BV
DOI: 10.1016/j.disc.2007.05.017
关键词
lattice; Delannoy numbers
类别
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.
作者
我是这篇论文的作者
点击您的名字以认领此论文并将其添加到您的个人资料中。
推荐
暂无数据