I'm curious as to how many matrices there are of size $m \times n$ with elements of the set $\{1, \ldots , k\}$ such that each row and column is weakly increasing?
The answer should be expressable as a determinant.
I'm thinking that this could be solved by counting non-intersecting lattice paths somehow and using Lindström–Gessel–Viennot lemma, however I'm unsure of how to construct the matrix.
Thanks!