Permanents of cyclic (0,1) matrices

Abstract An 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 form P n k = 2 + ∑ τ − 1 [ k − 1 2 ] T τ k ( 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 Menage Problem in terms of sums over scalar products of the iterates of a set of unit vectors.

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