Permanents of matrices of signed ones
By calculating the permanent 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
Minc'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.
Click here to download the whole paper.
Last modified: Tue Sep 7 19:10:34 EST 2004