Let n and t be integers with \(1<t\le n\) . Let A be an \(n\times n\) (0, 1)-matrix with a positive permanent, that is, for which there exists a permutation matrix \(P\le A\) (entrywise order). We investigate the minimum number \(\alpha (n,t)\) of zeros possible in such a matrix A which avoids a P with a \(12\cdots t\) -pattern, that is, for which there does not exist a permutation matrix \(P\le A\) containing the identity matrix \(I_t\) as a submatrix. We conjecture that \(\alpha (n,t)={{k+1}\atopwithdelims ()2}\) where \(k=n-t+1\) . We prove this conjecture is correct when \(t=2 \text{ or } 3\) and we consider for which matrices equality holds. We also prove the conjecture is correct for all t if \(n\ge 2k-3\) . Finally, we investigate which \(12\cdots t\) -permutation avoiding matrices have the maximum permanent.
