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Invariant sets for classes of matrices of zeros and ones


Authors: Richard A. Brualdi and Jeffrey A. Ross
Journal: Proc. Amer. Math. Soc. 80 (1980), 706-710
MSC: Primary 05B20
DOI: https://doi.org/10.1090/S0002-9939-1980-0587961-2
MathSciNet review: 587961
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Abstract: Let $ \mathfrak{A}(R,S)$ denote the class of all $ m \times n$ matrices of 0's and 1's with row sum vector R and column sum vector S. A set $ I \times J(I \subseteq \{ 1, \ldots ,m\} ,J \subseteq \{ 1, \ldots ,n\} )$ is said to be invariant if each matrix in $ \mathfrak{A}(R,S)$ contains the same number of 1's in the positions $ I \times J$. We prove that if there are no invariant singletons, then an invariant set $ I \times J$ satisfies $ I = \emptyset ,I = \{ 1, \ldots ,m\} ,J = \emptyset $ or $ J = \{ 1, \ldots ,n\} $.


References [Enhancements On Off] (What's this?)

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DOI: https://doi.org/10.1090/S0002-9939-1980-0587961-2
Article copyright: © Copyright 1980 American Mathematical Society

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