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Monotonicity conjecture on permanents of doubly stochastic matrices


Authors: Ko Wei Lih and Edward T. H. Wang
Journal: Proc. Amer. Math. Soc. 82 (1981), 173-178
MSC: Primary 15A15; Secondary 15A51
DOI: https://doi.org/10.1090/S0002-9939-1981-0609645-5
MathSciNet review: 609645
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Abstract: A stronger version of the van der Waerden permanent conjecture asserts that if $ {J_n}$ denotes the $ n \times n$ matrix all of whose entries are $ 1/n$ and $ A$ is any fixed matrix on the boundary of the set of $ n \times n$ doubly stochastic matrices, then $ {\text{per}}(\lambda A + (1 - \lambda ){J_n})$ as a function of $ \lambda $ is nondecreasing in the interval $ [0,1]$. In this paper, we elucidate the relation of this assertion to some other conjectures known to be stronger than van der Waerden's. We also show that this assertion is true when $ n = 3$ and in the case when, up to permutations of rows and columns, either (i) $ A = {J_s} \oplus {J_t}$, $ 0 < s$, $ t$, $ s + t = n$ or (ii) $ A = \left[\begin{smallmatrix}0 & Y \\ Y^T & Z\end{smallmatrix} \right]$ where 0 is an $ s \times s$ zero matrix, $ Y$ is $ s \times t$ with all entries equal to $ 1/t$, and $ Z$ is $ t \times t$ with all entries equal to $ (t - s)/{t^2}$, $ 0 < s \leqslant t$, $ s + t = n$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0609645-5
Keywords: Doubly stochastic matrix, permanent
Article copyright: © Copyright 1981 American Mathematical Society

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