Let \(A = a_1, a_2,\ldots, a_k\) and \(B = b_1, b_2, \ldots, b_j\) be sequences of positive integers such that \(a_1 \geq a_2 \geq \cdots a_k \geq 1,\) \(b_1 \geq b_2 \geq \cdots b_j \geq 1,\) \(\sum\limits_{i=1}^{k} a_i \leq 6\), and \(\sum\limits_{i=1}^{j} b_i \leq 6.\) 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 \(a_m\) and the sum of the column \(n\) of \(T\) is \(b_n\)?

**Details and assumptions**

\(j\) and \(k\) are not fixed values, and can be any number for which such sequences exist. As an explicit example, with \( A_1 = \{1\}, B_1 = \{1\}, A_2 = \{3, 3\}, B_2 = \{2,2,2\} \), then the pairs \( (A_1, B_1) \) and \( (A_2, B_2) \) are solutions, corresponding to \(1 \times 1 \) and \( 2 \times 3 \) tables filled with all 1's.

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