Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



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
MathSciNet review: 587961
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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?)

  • [1] R. A. Brualdi and J. A. Ross, On Ryser's maximum term rank formula, Linear Algebra and Appl. 29 (1980), 33-38. MR 562747 (81e:15008)
  • [2] D. Gale, A theorem on flows in networks, Pacific J. Math. 7 (1957), 1073-1082. MR 0091855 (19:1024a)
  • [3] H. J. Ryser, Combinatorial properties of matrices of zeros and ones, Canad. J. Math. 9 (1957), 371-377. MR 0087622 (19:379d)
  • [4] -, Combinatorial mathematics, The Carus Mathematical Monographs, No. 14, The Mathematical Association of America; distributed by Wiley, New York, 1963. MR 0150048 (27:51)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 05B20

Retrieve articles in all journals with MSC: 05B20

Additional Information

Article copyright: © Copyright 1980 American Mathematical Society

American Mathematical Society