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The volume of the 10th Birkhoff polytope

Matthias Beck·Dennis Pixton
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Abstract

The n'th Birkhoff polytope $B_n$ is the set of all doubly stochastic $n \times n$ matrices, that is, those matrices with nonnegative real coefficients in which every row and column sums to one. A long-standing open problem is the determination of the relative volume of $B_n$. In [arXiv:math.CO/0202267] we introduced a method of calculating this volume and used it to compute $\vol B_9$. This note is an update on our progress: with the same program (but much longer computing time), we have now derived $\vol B_{10}$.

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Matthias Beck, Dennis Pixton. 2003-05-23. The volume of the 10th Birkhoff polytope. https://arxiv.org/abs/math/0305332

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