The permanent of an $n$-by- $n$ matrix $A=\left(a_{i j}\right)$ is defined as $$ \operatorname{perm}(A)=\sum_{\sigma \in S_{n}} \prod_{i=1}^{n} a_{i, \sigma(i)} $$ The sum here extends over all elements $\sigma$ of the symmetric group $S_{n}$ i.e. over all permutations of the numbers $1,2, \ldots, n$. $$ \operatorname{perm}\left(\begin{array}{ll} a & b \\ c & d \end{array}\right)=a d+b c $$ The best known general exact algorithm of computing the permanent is due to H.J. Ryser. Ryser's formula can be evaluated using $\mathrm{O}\left(2^{n-1} n^{2}\right)$ arithmetic operations $$ \operatorname{perm}(A)=(-1)^{n} \sum_{S \subseteq\{1, \ldots, n\}}(-1)^{|S|} \prod_{i=1}^{n} \sum_{j \in S} a_{i j} $$
Is there any method or algorithm to detect a permanent of a square matrix $A(1,-1,0)$ with large even order(1000-1600) zero or nonzero by (super) computer? ( for instance $\left.A_{1000 \times 1000}, A_{1600 \times 1600} \cdot\right)$. It would be better if it could be calculated out by (super) computer. Thanks in advance.
