On extremal points of the space of symmetric stochastic matrices

Let $\Delta_n$ be the set of extremal points of the convex space of symmetric stochastic matrices of order $n$. Asymptotic formulas as $n\to\infty$ are found for the number of elements in $\Delta_n$, and limit distributions are given for the number of positive elements in a random probability matrix $C\in\Delta_n$ and for the characteristic multiplicity of a random mapping from a set of permutations in $\Delta_n$.
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