Permanents of matrices of signed ones

Abstract

By calculating the permanents for all Hadamard matrices of orders up to and including 28 we answer a problem posed by E.T.H. Wang and a similar question asked by H. Perfect. Both questions are answered by the existence of Hadamard matrices of order 20 which do not seem to be simply related but nevertheless have the same permanent. For orders up to and including 20 we also settle several other existence questions involving permanents of (+1, -1)-matrices. Specifically, we establish the lowest positive value taken by the permanent in these cases and find matrices which have equal permanent and determinant when such a matrix exists. Our results address Conjectures 19 and 36 and Problems 5 and 7 in Mine's well known catalogue of unsolved problems on permanents. We also include a little-known proof that there exists a (+1,-1)-matrix A of order n such that per(A) = 0 if and only if n+ 1 is not a power of 2.