Nonnegative matrices with zero permanent

We study $m\times n$ matrices, $m\ge n$, whose elements are either 1) arbitrary nonnegative numbers or 2) belong to a given finite set of nonnegative numbers that includes zero. In the finite case, we obtain an asymptotic expression, as $n\to\infty$, for the number of matrices with zero permanent. For any nonnegative matrix with zero permanent a standard representation is derived.