The volume of the 10th Birkhoff polytope

TL;DR

This paper computes the volume of the 10th Birkhoff polytope, advancing the understanding of these polytopes by extending previous calculations of their volumes.

Contribution

The authors successfully calculated the volume of the 10th Birkhoff polytope using an improved computational method, building on their prior work for B_9.

Findings

Computed the volume of B_10 for the first time.

Extended previous computational methods to larger polytopes.

Provided data that may inform future theoretical analysis.

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 $\vol B_{10}$.