A doubly stochastic matrix is a matrix A = (a_(i j)) such that a_(i j)>=0 and sum_i a_(i j) = sum_j a_(i j) = 1 is some field for all i and j. In other words, both the matrix itself and its transpose are stochastic. The following tables give the number of distinct doubly stochastic matrices (and distinct nonsingular doubly stochastic matrices) over Z_m for small m. m | doubly stochastic n×n matrices over Z_m 2 | 1, 2, 16, 512, ... 3 | 1, 3, 81, ... 4 | 1, 4, 256, ...
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