Doubly stochastic circulant matrices (Q1182741)

Let \(P_ n\) be the circulant matrix of order \(n\) with the first row of the form \((0,1,0,\ldots,0)\). Let \(G^ 2_ n\) denote the set of all \(n\times n\) doubly stochastic matrices of the form \(\alpha_ nI_ n+\beta P_ n+\gamma_ nP_ n^ 2\), and let \(\mu_ n\) denote the minimum value of the permanent on \(G^ 2_ n\). In the note the lower bound for \(\mu_ n\) obtained by Minc (1972) and Suchan (1981) is improved. The author also gives a rather simple formula for \[ \mu_ n=\min_{t\geq 1}(1/2^ n(1+t)^ n)\{(1+\sqrt{1+t^ 2})^ n+(1- \sqrt{1+t^ 2})^ n+2t^ n\}. \]