Question

Let A=a1,a2,…,ak and B=b1,b2,…,bj be sequences of positive integers such that a1≥a2≥⋯ak≥1, b1≥b2≥⋯bj≥1, ∑i=1k ai≤6, and ∑j i=1 bi≤6. For how many ordered pairs of sequences (A,B) satisfying the above conditions can we find a table T with {0,1} entries such that for each m,n, the sum of row m of T is am and the sum of the column n of T is bn?
Details and assumptions
j and k are not fixed values, and can be any number for which such sequences exist.