Given $m \times n$ rectangle with area $A$, and $m,n \in \mathbb{N}$. Let $S_k(m,n)$ be the number of way to dissect this rectangle into $k$ non-overlapping triangles whose area is $\frac{A}{k}$.
It is known that when $k$ is odd $S_k(n,n)=0$ (For example see Paul Monsky, "On Dividing A Square Into Triangles", American Mathematical Monthly, Vol 77 no 2, 1970)
Is it true that $S_k(m,n) < \infty$?
Is there any known bound for $S_k(m,n)$ (even when $m=n$ and $k$ even)?
Edit: There are several possible interpretations of the problem, one may assume that the vertices of the triangles are lattice points. Or perhaps (harder?), when we are given an arbitrary rectangle with a fixed dimension (not necessarily has an integer dimension).
In the case where the vertices of the triangles are not lattice points, there exists some $k$ for which $S_k(m,n)$ is not finite. In particular, $S_8(1,1)$ is infinite.
This result comes from the related problem of subdividing triangles into triangles. The number of ways to divide a triangle into 3 equal parts is finite; the number of ways to divide it into 4 equal parts is infinite. See here for reference: http://www.mathpuzzle.com/triangle.html