Maximum of per(I−A) for odd-order doubly stochastic matrices
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For odd n=2k+1>1 and doubly stochastic A, determine max per(I−A) and every equality case. The target is 3·2^(k−2), attained exactly by simultaneous permutation conjugates of one half-weight triangle block and k−1 swap blocks.
Why it matters and what counts as progress
Why it matters: It seeks a sharp odd-order matrix extremum together with the full structure of maximizers.
What counts as progress: Establish the sharp odd-order permanent maximum and classify every attaining doubly stochastic matrix.
How the problem developed
- 1981 · Conjecture [1]
Jin H. Kim and James G. Roush Conjectured the maximum permanent of I−A for doubly stochastic matrices of odd order.
- 9 Aug 2026 · Proof [1]
Yair Lavi Proved the conjectured maximum 3·2^(k−2) for order 2k+1 and characterized the equality cases up to simultaneous permutation conjugacy.
AI contribution
The Maximum of per(I−A) in Odd Order
For odd n=2k+1>1 and doubly stochastic A, Lavi proves max per(I−A)=3·2^(k−2). Every maximizer is a simultaneous permutation conjugate of the half-weight triangle block plus k−1 swap blocks.
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