We wish to consider the class of (0,1)-matrices with prescribed row and column sums. Let R = (r1, r2, …, rm) and S= (s1, s2, …, sn) be vectors with nonnegative integral entries and r1 + r2 + … + rm = s1 + s2 + … + sn. We define the class
to be the set of m × n (0, 1)-matrices with ith row sum ri and jth column sum sj for 1 ≦ i ≦ m and 1 ≦ j ≦ n.
Gale and Ryser independently found simple necessary and sufficient conditions for
to be nonempty [9 , 14]. From R, we form an m × n (0, 1)-matrix Ā as follows. The ith row sum of Ā is ri and the 1‘s are as far to the left as possible. Let
be the jth column sum of A. We define the sequence
to be the conjugate of the sequence (ri).