Computational evidence supports a Boundary Conjecture that pairs of permutation matrices determine the boundary of the doubly stochastic eigenvalue region, and indicates that the Perfect-Mirsky region equals the full region for n=6 through 11, with a unique exception at n=5.
Bouc, Burnside rings , in Handbook of Algebra, vol
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The Doubly Stochastic Single Eigenvalue Problem: A Computational Approach
Computational evidence supports a Boundary Conjecture that pairs of permutation matrices determine the boundary of the doubly stochastic eigenvalue region, and indicates that the Perfect-Mirsky region equals the full region for n=6 through 11, with a unique exception at n=5.