Triangular blocks of zeros in (0,1) matrices with small permanents (Q676040)

If \(A\) is a square matrix and \(t\) a positive integer, \(A\) is said to be \(t\)-triangular if there exist permutation matrices \(P\) and \(Q\) such that \(PAQ=B=[b_{ij}]\) has \(b_{ij} =0\) whenever \(j\geq i+t\). The truth of the following statement is investigated: If \(A\) is any square matrix with nonnegative integral entries such that \(0< \text{per} A< (t+1)!\), then \(A\) is \(t\)-triangular. If \(t=1\) the statement reduces to a theorem of Brualdi. It is proved here that the statement is true for \(t=2\) and \(t=3\), but false for \(t=6\).