Some Conjectures on Permanents of Doubly Stochastic Matrices

Conjectures on permanents are well-known unsettled conjectures in linear algebra. Let A be an n×n matrix and Sn be the symmetric group on n element set. The permanent of A is defined as perA=∑σ∈Sn∏i=1naiσ(i). The Merris conjectured that for all n×n doubly stochastic matrices (denoted by Ωn), nperA≥min1≤i≤n∑j=1nperA(j|i), where A(j|i) denotes the matrix obtained from A by deleting the jth row and ith column. Foregger raised a question whether per(tJn+(1−t)A)≤perA for 0≤t≤nn−1 and for all A∈Ωn, where Jn is a doubly stochastic matrix with each entry 1n. The Merris conjecture is one of the well-known conjectures on permanents. This conjecture is still open for n≥4. In this paper, we prove the Merris inequality for some classes of matrices. We use the sub permanent inequalities to prove our results. Foregger’s inequality is also one of the well-known inequalities on permanents, and it is not yet proved for n≥5. Using the concepts of elementary symmetric function and subpermanents, we prove the Foregger’s inequality for n=5 in [0.25, 0.6248]. Let σk(A) be the sum of all subpermanents of order k. Holens and Dokovic proposed a conjecture (Holen–Dokovic conjecture), which states that if A∈Ωn,A≠Jn and k is an integer, 1≤k≤n, then σk(A)≥(n−k+1)2nkσk−1(A). In this paper, we disprove the conjecture for n=k=4.

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“…There is not much progress in this conjecture. Subramanian and Somasundaram [8] have proved that if A ∈ Ω n and the polynomial…”

Section: Merris Conjecturementioning

confidence: 99%

“…The conjecture is still open for n ≥ 4. Subramanian and Somasundaram [8] have proved that if A ∈ Ω n and the polynomial n ∑ r=2 r (n−r)! n n−r σ r (A − J n )t r−2 has no root in (0, 1) then A satisfies Merris conjecture.…”

Section: Introductionmentioning

confidence: 99%

The Inequalities of Merris and Foregger for Permanents

Udayan

Somasundaram

2021

Symmetry

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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 Conjectures on Permanents of Doubly Stochastic Matrices

Cited by 2 publications

References 11 publications

The Inequalities of Merris and Foregger for Permanents

The Inequalities of Merris and Foregger for Permanents

Lih Wang's and Dittert's conjectures on permanents

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