Counting lattice chains and Delannoy paths in higher dimensions

Lattice chains and Delannoy paths represent two different ways to progress through a lattice. We use elementary combinatorial arguments to derive new expressions for the number of chains and the number of Delannoy paths in a lattice of arbitrary finite dimension. Specifically, fix nonnegative integers n 1 , … , n d , and let L denote the lattice of points ( a 1 , … , a d ) ∈ Z d that satisfy 0 ≤ a i ≤ n i for 1 ≤ i ≤ d . We prove that the number of chains in L is given by 2 n d + 1 ∑ k = 1 k max ′ ∑ i = 1 k ( − 1 ) i + k k − 1 i − 1 n d + k − 1 n d ∏ j = 1 d − 1 n j + i − 1 n j , where k max ′ = n 1 + ⋯ + n d − 1 + 1 . We also show that the number of Delannoy paths in L equals ∑ k = 1 k max ′ ∑ i = 1 k ( − 1 ) i + k ( k − 1 i − 1 ) ( n d + k − 1 n d ) ∏ j = 1 d − 1 ( n d + i − 1 n j ) . Setting n i = n (for all i ) in these expressions yields a new proof of a recent result of Duchi and Sulanke [9] relating the total number of chains to the central Delannoy numbers. We also give a novel derivation of the generating functions for these numbers in arbitrary dimension.