Let $X_r\subset Mat_{n\times n}$ denote the matrices of rank at most
$r$, and let $S_{\pi}C^n$ denote the irreducible $GL_n$-module corresponding
to the partition $\pi$. 
One can check that
degree($X_r$)=dim($S_{(n-r)^{n-r}}C^n)$=$\Pi_{i=0}^{n-r-1}\frac{(n+i)!i!}{(r+i)!(n-r+i)!}$
 Does anyone have a geometric (or any) explanation for this equality? Had it been observed previously?
Here $(n-r)^{n-r}=(n-r,...,n-r)$ has Young diagram the square box.