  Let $\Omega_n$ 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 $\Omega_n$. The main question is: which
$A \in \Omega_n$ 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.
