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The Foregger–Sinkhorn Tie-Point Conjecture

The Foregger–Sinkhorn tie-point conjecture, Conjecture 41 in Minc's survey, asserts that if a nearly decomposable doubly stochastic matrix minimizes the permanent on a face and the permanental cofactor at a prescribed zero exceeds its permanent, then that zero is a tie point. False: an explicit $8 \times 8$ counterexample exists, built on the unique root $\beta$ of $7t^3 - 13t^2 + 12t - 4$ in $(59/100, 3/5)$.

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combinatoricsAug 13, 2026Significance 12/100Registry: unreviewed

The Foregger–Sinkhorn Tie-Point Conjecture

Prior state unknown→disproved

The Foregger–Sinkhorn tie-point conjecture, Conjecture 41 in Minc's survey, asserts that if a nearly decomposable doubly stochastic matrix minimizes the permanent on a face and the permanental cofactor at a prescribed zero exceeds its permanent, then that zero is a tie point. False: an explicit $8 \times 8$ counterexample exists, built on the unique root $\beta$ of $7t^3 - 13t^2 + 12t - 4$ in $(59/100, 3/5)$.

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The Foregger–Sinkhorn tie-point conjecture, Conjecture 41 in Minc's survey, asserts that if a nearly decomposable doubly stochastic matrix minimizes the permanent on a face and the permanental cofactor at a prescribed zero exceeds its permanent, then that zero is a tie point. False: an explicit $8 \times 8$ counterexample exists, built on the unique root $\beta$ of $7t^3 - 13t^2 + 12t - 4$ in $(59/100, 3/5)$.

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The Foregger–Sinkhorn Tie-Point Conjecture — Mathematical Frontier Network