Monotonicity of permanents of direct sums of doubly stochastic matrices |
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Authors: | John Goldwasser |
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Institution: | Department of Mathematics , West Virginia University , Morgantown, WV, 26506 |
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Abstract: | Let Ωn be the set of all n × n doubly stochastic matrices, let Jn be the n × n matrix all of whose entries are 1/n and let σ k (A) denote the sum of the permanent of all k × k submatrices of A. It has been conjectured that if A ε Ω n and A ≠ JJ then gA,k (θ) ? σ k ((1 θ)Jn 1 θA) is strictly increasing on 0,1] for k = 2,3,…,n. We show that if A = A 1 ⊕ ⊕At (t ≥ 2) is an n × n matrix where Ai for i = 1,2, …,t, and if for each i gAi,ki (θ) is non-decreasing on 0.1] for kt = 2,3,…,ni , then gA,k (θ) is strictly increasing on 0,1] for k = 2,3,…,n. |
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