A conjecture on permanents (Q1894470)

The conjecture on permanents by \textit{R. A. Brualdi} [Linear Multilinear Algebra 17, 5-18 (1985; Zbl 0564.15010)] that the \(n\times n\) \((0,1)\) matrix with the last \(n-1\) entries on the main diagonal equal to 0 and all the other entries equal to 1 is never barycentric for \(n\geq 4\) is proved (the barycenter is defined as \(b(D)={1\over\text{per} D}\sum_{p\leq D}P\), where \(D\) is an \(n\times n\) \((0,1)\) matrix). Three cases are distinguished in the proof: \(n=4\), \(n\) is any even integer greater than 4, and \(n\) is any odd integer greater than 4.