Solutions of permanental equations regarding stochastic matrices (Q1586268)

Let \(A\) be an \(n\times n\) doubly stochastic matrix, \(1\leq k\leq n\). For \(1\leq i_1< i_2<\dots <i_k\leq n\) and \(1\leq j_1<j_2<\dots <j_k\leq n\), let \(A_{i_1\dots i_k}^{j_1\dots j_k}\) be the \(k\times k\) matrix consisting of the entries on the \(i_1\)th, \(i_2\)th, \(\dots\), \(i_k\)th rows and the \(j_1\)th, \(j_2\)th, \(\dots\), \(j_k\)th columns of \(A\). Define \(t_0(A)=1\), and \[ t_k(A)=\frac {\sum_{i_1<\dots <i_k, j_1<\dots <j_k} \text{Permanent}(A_{i_1\dots i_k}^{j_1\dots j_k})}{\binom nk^2}. \] The paper shows, by means of the above operators and concominants, that if the upper and lower permanents of the product of two \(n\times n\) doubly stochastic matrices are equal then at least one factor is the matrix with all entries equal to \(\frac 1n\). In addition, a few results about the means of permanents of doubly stochastic matrices are obtained.