UPPER BOUNDS FOR SUBPERMANENTS OF NONNEGATIVE MATRICES

For an $n \times n$ matrix $A = [a_{ij}]$, the permanent of A, per A, is defined by $$ per(A) = \sum_{\sigma}{a_{1 \simga(1)} \cdots a_{n \sigma(n)}}, $$ where $\sigma$ runs over all permutations of ${1,\cdots,n}.