On the term rank partitions of matrices in A(R, S)

The weight of a partition is the sum of its nonzero coordinates. Let R and S be partitions of the same weight and (Formula presented.) be the class of all (Formula presented.) -matrices with row sums R and column sums S. For a positive integer t, the t-term rank of a matrix (Formula presented.), denoted (Formula presented.), is defined as the largest number of 1's in A with at most one 1 in each column and at most t 1's in each row. The term rank partition of A is the partition (Formula presented.) where (Formula presented.). Let θ be a partition of weight equal to the number of nonzero coordinates of S. In this paper, we study conditions for the existence of a matrix (Formula presented.) with (Formula presented.).