# Sequences and Tables

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.

×