A note on lattice chains and Delannoy numbers

Fix nonnegative integers n1,…,ndn1,…,nd and let L   denote the lattice of integer points (a1,…,ad)∈Zd(a1,…,ad)∈Zd satisfying 0⩽ai⩽ni0⩽ai⩽ni for 1⩽i⩽d1⩽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=2d=2. Setting n1=n2n1=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.