Minc’s list

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Research project · Updated September 26, 2026

Minc’s list

This is a list of the problems and conjectures from Minc’s list, Henryk Minc’s catalogue of open questions on permanents, that were still open when we started working on them in late July 2026.

For a survey of the problems and conjectures that were settled before that date, see G.-S. Cheon and I. M. Wanless, An update on Minc’s survey of open problems involving permanents, Linear Algebra and its Applications 403 (2005), 314–342 (free PDF). However, notice that Conjectures 31, 32, 37, 38 and 40, and Problem 6, were settled in the years between that survey and the start of our project.

Open in July 2026
34
Resolved by us
5
Resolved by others
1
Still open
28
Number Name Originator Date Status Partial results
Conjectures 19
Conjecture 3 Marcus–Minc transform inequality Marcus and Minc 1967 Proved by us Proved in every order n ≥ 2, with all equality cases.
Conjecture 4 Wang transform inequality Wang 1977 Proved by us Proved in every order, with equality only at Jₙ.
Conjecture 5 Ryser’s design conjecture Ryser 1960 Open Known for v ≤ 12; at v = 13 the designs beat every matrix within two switches.
Conjecture 6 Monotonicity of normalized minimum permanents Minc 1965 Open Known for v ≤ 11 and in sparse and dense ranges; we proved two lines in every order.
Conjecture 9 Marcus’s block-permanent inequality Marcus 1965 Open Known for two blocks; we proved three 2×2 blocks at rank ≤ 2, and reduced the rest to ranks 3–4.
Conjecture 15 Foregger’s nearly decomposable minimum Foregger 1978 Open Bound known for n ≤ 9; we proved it on infinite pattern families, one with the equality case.
Conjecture 17 Foregger’s powers conjecture Foregger 1978 Proved by others Proved: per(Aᵏ) ≤ per A for every k ≥ K(n); we found an independent proof.
Conjecture 18 Merris’s average-minor inequality Merris 1973 Open Known for positive semidefinite A; we proved it for circulant, block and sparse families in every order.
Conjecture 20 Gyires’s ratio inequality Gyires 1978 Open Known for n ≤ 3; we proved it on the whole boundary of Ω₄ and near Jₙ.
Conjecture 24 Integrality of the minimum permanent Minc 1983 Open We proved it, with every minimizer a (0,1)-matrix, for all n ≤ 7 and for (n, k) = (8, 3).
Conjecture 27 Few permanents of prime circulants Nemeth, Seberry and Shu 1979 Open True for row sums ≤ 3; we found exact counts for row sum 4 up to p = 43, and more than 53 values at p = 53.
Conjecture 28 Dittert’s conjecture Dittert 1983 Open Known for n ≤ 4 and n ≥ 16; we proved 8 ≤ n ≤ 15, leaving only n = 5, 6, 7.
Conjecture 30 Chollet’s conjecture Chollet 1982 Open Known for n ≤ 6 (2026 preprint); we proved rank ≤ 2 in every order, and rank ≤ 3 except n = 7.
Conjecture 34 Lih–Wang convexity inequality Lih and Wang 1982 Open Known for n = 3; we proved n = 4 in full, and n = 5 for mixtures of four permutations.
Conjecture 35 Kim–Roush odd-order maximum Kim and Roush 1981 Proved by us Proved in every odd order, with all maximizers classified.
Conjecture 36 Kräuter’s minimum sign-matrix permanent Kräuter 1985 Open Known for n ≤ 20; we constructed attaining matrices for 21 ≤ n ≤ 32.
Conjecture 41 Foregger–Sinkhorn tie-point conjecture Foregger and Sinkhorn 1986 Disproved by us False: we found an exact counterexample of order 8.
Conjecture 42 Lieb’s permanental dominance conjecture Lieb 1966 Open We proved every immanant case for n ≤ 17; the general form is known for n ≤ 4 (2026 preprint).
Conjecture 44 Zero-diagonal doubly stochastic minimum Folklore 1987 Open Known for n ≤ 4; we confined counterexamples near (J − I)/(n − 1) in every order.
Problems 15
Problem 1 Maximum permanent on a unitary orbit Marcus and Minc 1965 Open Solved for n ≤ 3; we found necessary conditions in every order and an explicit lower bound for n = 4.
Problem 2 Permanent dominance for degree-one characters Marcus and Minc 1965 Open Known for n ≤ 3; we proved every case for n = 4 and 38 of the 45 cases for n = 5.
Problem 3 Permanent values of 3-regular matrices Greenstein 1978 Open Value sets known for n ≤ 11; we proved the number of values grows exponentially.
Problem 4 Maximum permanent when k ∤ n Minc 1978 Open Solved for k ≤ 3; we settled the line k = n − 3 for every n ≥ 69.
Problem 5 Vanishing Hadamard permanents Wang 1974 Open Nonzero for orders below 32; we proved it for every Hadamard matrix of order below 128.
Problem 7 Sign matrices with |per A| = |det A| Wang 1974 Open Answer known for n ≤ 20; our explicit matrices settle 21 ≤ n ≤ 25.
Problem 8 Monotonicity toward the boundary Friedland and Minc 1978 Open Many monotone classes known; we found non-monotone examples in every order n ≥ 9.
Problem 10 Exponential bounds for 6-regular circulants Minc 1978 Solved by us Solved: we found the optimal circulant constant h₆, with 2.41133 ≤ h₆ ≤ 2.54756.
Problem 11 Matrices below every circulant Minc 1983 Open Answered for n ≤ 11; we answered it for large n with k fixed, and in new dense and cubic cases.
Problem 12 Circulant maxima when k ∤ n Minc 1983 Open Answered for n ≤ 11; we proved an explicit circulant bound and effective thresholds for part two.
Problem 13 Real van der Waerden radius Sinkhorn 1981 Open We found the exact values b(3) = 3√2/4 and b(4) ≈ 1.16145, and bounds for n ≥ 5.
Problem 14 Characterizing cohesive matrices Brualdi 1985 Open Many families known; we settled order 3 and reduced the question to irreducible cores.
Problem 15 Characterizing barycentric matrices Brualdi 1985 Open Many classes known; we solved simplex faces and found new infinite non-barycentric families.
Problem 17 Characteristic polynomial of permanental compounds Minc 1986 Open Solved for special companion matrices; we found faster exact methods for k = 2, low rank and block matrices.
Problem 18 Perron root of permanental compounds Minc 1986 Open Bounds known; we found exact methods for low rank and k = 2, and fast approximation for broad classes.
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Numbering follows Minc’s catalogue as retained by Cheon and Wanless. Select a number for the exact statement, its history, and our progress.