Doubly stochastic circulant matrices

AbstractLet Pn denote the n×n permutation matrix with 1's in the positions (i,j) where j ≡ i + 1 (mod n). Let Γ2n denote the set of all n x n doubly stochastic circulants of the form αIn + βPn + γP2n, and let μn denote the minimum value of the permanent on Γ2n. Minc (1972) and Suchan (1981) proved that 2-n < μn < 21−n. This note proves the following assertions: 1.(i) If aIn + bPn + cP2n ϵ Γ2n is a matrix of minimum permanent on Γ2n, then 0 < b < 12, 14 < a = c < 12;2.(ii) μn = mint⩾1 (12n(1+t)n){(1 + √1+t2)n + (1 − √1+t2)n + 2tn};3.(iii) 2-n + 2-2n < μn < 21−n