On the t-term rank of a matrix
For t a positive integer, the t-term rank of a (0,1)-matrix A is defined to be the largest number of 1s in A with at most one 1 in each column and at most t 1s in each row. Thus the 1-term rank is the ordinary term rank. We generalize some basic results for the term rank to the t-term rank, including a formula for the maximum term rank over a nonempty class of (0,1)-matrices with the same row sum and column sum vectors. We also show the surprising result that in such a class there exists a matrix which realizes all of the maximum terms ranks between 1 and t.
Brualdi RA, Kiernan KP, Meyer SA, Schroeder MW. On the t-term rank of a matrix. Linear Algebra and its Applications. 2012;436(6):1632-43.