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Dittert Conjecture

The Dittert conjecture is an inequality for the permanent of a nonnegative n×n matrix whose entries sum to n. If r_i and c_j are its row and column sums, it asserts that

product_(i=1)^nr_i+product_(j=1)^nc_j-per(A)<=2-(n!)/(n^n),

with equality only when all entries equal 1/n, for n>=2 (Dai et al. 2026). For a doubly stochastic matrix, this reduces to per(A)>=n!/n^n.

Dai et al. (2026) reported a proof using boundary rigidity. The complete proof has not been independently verified here. Dai et al. (2026) state that the TARS agent system assisted exploratory mathematical search and derivations. Xinan Dai reconstructed the proof, supplied omitted details, and manually verified the mathematical results.

See also

Doubly Stochastic Matrix, Permanent

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References

Dai, X.; Deng, W.; Shi, Y.; Wu, T.; and Yang, Y. "A Proof of the Dittert Conjecture via Boundary Rigidity." 30 Jul 2026. https://arxiv.org/abs/2610.10545.

Cite this as:

Weisstein, Eric W. "Dittert Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DittertConjecture.html

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