I want to count the number of shortest paths from one corner of a multidimensional lattice, to the opposite corner. Hence in the 2-dimensional case, given a lattice (or grid graph) of $n_1 \times n_2$ vertices, I want to count the number of shortest paths from the top left vertex to the bottom right vertex (which must only move right and down). In this case, the number of such paths is given by $\binom{(n_1-1)+(n_2-1)}{n_1-1}$
which for $n_1=3$ and $n_2=2$ equals 3 (see Staircase Walks). Now I want to find the following: Given integers $(n_1,n_2,\ldots,n_k)$, what is the number of shortest paths from one corner of the $(n_1 \times n_2 \times \cdots \times n_k)$-dimensional lattice to the opposite?