Abstract
An efficient method is presented for evaluating the permanents P n k 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 P n k ( k fixed); the recurrence rules are given in semiexplicit form for the range 4≤ k≤9. A table of P n k is included for the range 4≤ k≤9, k≤ n≤80. The P n k are calculated in the form P n k = 2 + ∑ τ − 1 [ k − 1 2 ] T τ k ( n ) where the T t k(n) satisfy recurrence rules given symbolically by the characteristic equations of certain (0, 1) matrices Π r k ; the latter turn out to be identical with the r-th permanental compounds of certain simpler matrices Π 1 k . Finally, formal expressions for P n k 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.
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