Permanents of matrices of signed ones

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 (þ1, �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 (þ1, �1)-matrix A of order n such that perðAÞ ¼ 0 if and only if n þ 1 is not a power of 2

Permanents of cyclic (0,1) matrices

AbstractAn efficient method is presented for evaluating the permanents Pnk of cyclic (0,1) matrices of dimension n and common row and column sum k. A general method is developed for finding recurrence rules for Pnk (k fixed); the recurrence rules are given in semiexplicit form for the range 4≤k≤9. A table of Pnk is included for the range 4≤k≤9, k≤n≤80. The Pnk are calculated in the formPnk=2+∑τ−1[k−12]Tτk(n)where the Ttk(n) satisfy recurrence rules given symbolically by the characteristic equations of certain (0, 1) matrices Πrk; the latter turn out to be identical with the r-th permanental compounds of certain simpler matrices Π1k. Finally, formal expressions for Pnk are given which allow one to write down the solution to the generalized Ménage Problem in terms of sums over scalar products of the iterates of a set of unit vectors

On permanents of Sylvester Hadamard matrices

It is well-known that the evaluation of the permanent of an arbitrary $(-1,1)$-matrix is a formidable problem. Ryser's formula is one of the fastest known general algorithms for computing permanents. In this paper, Ryser's formula has been rewritten for the special case of Sylvester Hadamard matrices by using their cocyclic construction. The rewritten formula presents an important reduction in the number of sets of $r$ distinct rows of the matrix to be considered. However, the algorithm needs a preprocessing part which remains time-consuming in general