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.

中文翻译：

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双随机矩阵的对角线和

让${\mathrm{欧姆}}_n$表示类$n\times n$双随机矩阵（每个这样的矩阵都是逐项非负的，并且每一行和每一列的总和都是 1）。我们研究矩阵的对角线${\mathrm{欧姆}}_n$. 主要问题是$\mathrm{一种}\in {\mathrm{欧姆}}_n$是这样的，即A中避免 A 的零点的对角线都具有相同的条目之和。我们给出了此类矩阵的特征，并建立了此类矩阵的几类模式。