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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the monotonicity of the permanent

Author: Suk Geun Hwang
Journal: Proc. Amer. Math. Soc. 106 (1989), 59-63
MSC: Primary 15A15; Secondary 15A51
MathSciNet review: 960645
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Abstract: Let $ {\Omega _n}$ denote the set of all $ n \times n$ doubly stochastic matrices and let $ {J_n} = {[1/n]_{n \times n}}$. For $ A \in {\Omega _n}$, if $ {f_A}(t) = {\text{per((1 - t)}}{J_n} + tA)$ is a nondecreasing function of $ t$ on $ [0,1]$, we say that the monotonicity of permanent (abb. MP) holds for $ A$ . Friedland and Mine [3] proved MP for $ (n{J_n} - {I_n})/(n - 1)$. In [6], Lih and Wang proposed a problem of determining whether MP holds for $ {J_{{n_1}}} \otimes \cdots \otimes {J_{{n_k}}},{n_i} > 0$.

In this note, we prove MP for $ ((m{J_m} - {I_m}) \otimes s{J_s})/(m - 1)s$, extending the result of Friedland and Mine, and give an affirmative answer to the Lih and Wang's question.

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Keywords: Permanent, doubly stochastic matrix, monotonicity
Article copyright: © Copyright 1989 American Mathematical Society

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