Permanents of cyclic (0,1) matrices

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$P_n^k=2+\sum _{\tau -1}^{\left[\frac{k-1}{2}\right]}{T}_{\tau }^k\left(n\right)$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 Π₁^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.