Invariant sets for classes of matrices of zeros and ones

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 $J = \{ 1, \ldots ,n\}$.