HMMT 2006 T3

In this problem, we stipulate that $m \geq n$, and the lawn is shaped differently. The clumps are now at the lattice points in a trapezoid:

$$\left\{(x, y) \in \mathbb{Z}^{2} \mid 0 \leq x<n \text { and } 0 \leq y<m+1-n+x\right\}$$

As in problem 2, mobots can be set to move either north or east. For given $m$ and $n$, determine with proof the number of allowable formations.