Some results towards the Dittert conjecture on permanents

Cheon, Gi-Sang; Wanless, Ian M.

doi:10.1016/j.laa.2010.08.041

Cited by 3 publications

(3 citation statements)

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“…Hwang further showed in [29] that if the ψ-maximising matrix is positive then it must equal J n , and that J n is a strict local maximum of ψ. Other partial results were obtained by Cheon and Yoon [9] and Cheon and Wanless [8]. Most importantly Cheon and Wanless [8] showed that the maximum value of ψ is exponentially close to the conjectured value.…”

Section: A Permanent Détourmentioning

confidence: 62%

“…Other partial results were obtained by Cheon and Yoon [9] and Cheon and Wanless [8]. Most importantly Cheon and Wanless [8] showed that the maximum value of ψ is exponentially close to the conjectured value.…”

Section: A Permanent Détourmentioning

confidence: 62%

“…Theorem 39 (Cheon-Wanless, [8]). For any non-negative n × n matrix A with the sum of all elements equal to 1 we have ψ(A) < ψ(J n ) + O(n 4−n e 2n ).…”

Section: A Permanent Détourmentioning

confidence: 99%

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On the algebraic and topological structure of the set of Turán densities

Grosu

2016

Journal of Combinatorial Theory, Series B

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The present paper is concerned with the various algebraic structures supported by the set of Turán densities.We prove that the set of Turán densities of finite families of r-graphs is a non-trivial commutative semigroup, and as a consequence we construct explicit irrational densities for any r ≥ 3. The proof relies on a technique recently developed by Pikhurko.We also show that the set of all Turán densities forms a graded ring, and from this we obtain a short proof of a theorem of Peng on jumps of hypergraphs.Finally, we prove that the set of Turán densities of families of r-graphs has positive Lebesgue measure if and only if it contains an open interval. This is a simple consequence of Steinhaus's theorem.

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Section: A Permanent Détourmentioning

confidence: 62%

Section: A Permanent Détourmentioning

confidence: 62%

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On the algebraic and topological structure of the set of Turán densities

Grosu

2016

Journal of Combinatorial Theory, Series B

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Permanents

Wanless¹

2013

Discrete Mathematics and Its Applications

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Lih Wang's and Dittert's conjectures on permanents

Udayan,

Somasundaram

2024

Special Matrices

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Let Ω n {\Omega }_{n} denote the set of all doubly stochastic matrices of order n n . Lih and Wang conjectured that for n ≥ 3 n\ge 3 , per ( t J n + ( 1 − t ) A ) ≤ t \left(t{J}_{n}+\left(1-t)A)\le t per J n + ( 1 − t ) {J}_{n}+\left(1-t) per A A , for all A ∈ Ω n A\in {\Omega }_{n} and all t ∈ [ 0.5 , 1 ] t\in \left[0.5,1] , where J n {J}_{n} is the n × n n\times n matrix with each entry equal to 1 n \frac{1}{n} . This conjecture was proved partially for n ≤ 5 n\le 5 . Let K n {K}_{n} denote the set of nonnegative n × n n\times n matrices whose elements have sum n n . Let ϕ \phi be a real valued function defined on K n {K}_{n} by ϕ ( X ) = ∏ i = 1 n r i + ∏ j = 1 n c j \phi \left(X)={\prod }_{i=1}^{n}{r}_{i}+{\prod }_{j=1}^{n}{c}_{j} - per X X for X ∈ K n X\in {K}_{n} with row sum vector ( r 1 , r 2 , … r n ) \left({r}_{1},{r}_{2},\ldots {r}_{n}) and column sum vector ( c 1 , c 2 , … c n ) \left({c}_{1},{c}_{2},\ldots {c}_{n}) . A matrix A ∈ K n A\in {K}_{n} is called a ϕ \phi -maximizing matrix if ϕ ( A ) ≥ ϕ ( X ) \phi \left(A)\ge \phi \left(X) for all X ∈ K n X\in {K}_{n} . Dittert conjectured that J n {J}_{n} is the unique ϕ \phi -maximizing matrix on K n {K}_{n} . Sinkhorn proved the conjecture for n = 2 n=2 and Hwang proved it for n = 3 n=3 . In this article, we prove the Lih and Wang partially for n = 6 n=6 . It is also proved that if A A is a ϕ \phi -maximizing matrix on K 4 {K}_{4} , then A A is fully indecomposable.

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Some results towards the Dittert conjecture on permanents

Cited by 3 publications

References 11 publications

On the algebraic and topological structure of the set of Turán densities

On the algebraic and topological structure of the set of Turán densities

Permanents

Lih Wang's and Dittert's conjectures on permanents

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