Invariant sets for classes of matrices of zeros and ones

ABsTmAcr. Let 2((R, S) denote the class of all m x n matrices of O's and l's with row sum vector R and column sum vector S. A set I X J (I c {1, . . J C { 1, . . ., n}) is said to be invariant if each matrix in W(R, S) contains the same number of l's in the positions I X J. We prove that if there are no invariant singletons, then an invariant set I x J satisfies I = 0, I -{1 (.. . m}, J 0 or J {l,.. ).