Possible Numbers of the 0-1 Rectangular Matrices Whose Rank Is 1 or 2

We have the possible numbers of ones in a 0-1 matrix with given rank in the generic case and in the symmetric case. There are some unexpected phenomena. The rank 2 symmetric case is subtle. On this basis, we carry out the promotion of literature, from the square matrix to a rectangular matrix {0,1}(mxn) with rank for the number of 1 or 2.The main conclusions: (1) sigma(1)(m, n) = (2(m) - 1)(2(n) - 1), (2) sigma(2)(m, n) = 4(n-2) n(n - 1)Sigma(m-l)(k=1)C(m)(k)(Sigma(m-k)(l=1)C(m-k)(l)) + 3(n-2)n(n - 1)(2(m) - 1)(2).