Diagonal Sums of Doubly Stochastic Matrices

Let ${\mathrm{\Omega }}_n$ denote the class of $n\times n$ doubly stochastic matrices (each such matrix is entrywise nonnegative and every row and column sum is 1). We study the diagonals of matrices in ${\mathrm{\Omega }}_n$. The main question is: which $A\in {\mathrm{\Omega }}_n$ are such that the diagonals in $A$ that avoid the zeros of $A$ all have the same sum of their entries. We give a characterization of such matrices, and establish several classes of patterns of such matrices.