On a question of Erdős on doubly stochastic matrices (Q6649803)

In 1959 \textit{M. Marcus} and \textit{R. Ree} [Q. J. Math., Oxf. II. Ser. 10, 296-302 (1959; Zbl 0138.01501)] showed that if \(A=[a_{ij}]\) is a doubly stochastic \(n \times n\) matrix, then there exists a permutation \(\sigma \in S_n\) such that such that \(\sum_{i,j=1}^n a_{i,j}^2 \leq \sum_{i=1}^n a_{i, \sigma (i) } \). It seems that Erdős asked for the characterization of doubly stochastic matrices for which the inequality is saturated. In this paper, the authors answer this question for all \(n \leq 3\).