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The famous Birkhoff-von Neumann theorem asserts that every doubly stochastic matrix can be written as a convex combination of permutation matrices.

The question is to point out different generalizations of this theorem, different "non-generalizations" namely cases where an expected generalization is false, and to briefly describe the context of these generalizations.

A related MO question: http://mathoverflow.net/questions/73805/sampling-from-the-birkhoff-polytope

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The famous Birkhoff-von Neumann theorem asserts that every doubly stochastic matrix can be written as a convex combination of permutation matrices.

The question is to point out different generalization generalizations of this theorem, different "non-generalizations" namely cases where an expected generalization is false, and to briefly describe the context of these generalizations.

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Generalizations of the Birkhoff-von Neumann Theorem

The famous Birkhoff-von Neumann theorem asserts that every doubly stochastic matrix can be written as a convex combination of permutation matrices.

The question is to point out different generalization of this theorem, different "non-generalizations" namely cases where an expected generalization is false, and to briefly describe the context of these generalizations.