Sequences and Tables

Probability Level 5

Let A=a1,a2,,akA = a_1, a_2,\ldots, a_k and B=b1,b2,,bjB = b_1, b_2, \ldots, b_j be sequences of positive integers such that a1a2ak1,a_1 \geq a_2 \geq \cdots a_k \geq 1, b1b2bj1,b_1 \geq b_2 \geq \cdots b_j \geq 1, i=1kai6\sum\limits_{i=1}^{k} a_i \leq 6, and i=1jbi6.\sum\limits_{i=1}^{j} b_i \leq 6. For how many ordered pairs of sequences (A,B)(A,B) satisfying the above conditions can we find a table TT with {0,1}\{0,1\} entries such that for each m,n,m,n, the sum of row mm of TT is ama_m and the sum of the column nn of TT is bnb_n?

Details and assumptions

jj and kk are not fixed values, and can be any number for which such sequences exist. As an explicit example, with A1={1},B1={1},A2={3,3},B2={2,2,2} A_1 = \{1\}, B_1 = \{1\}, A_2 = \{3, 3\}, B_2 = \{2,2,2\} , then the pairs (A1,B1) (A_1, B_1) and (A2,B2) (A_2, B_2) are solutions, corresponding to 1×11 \times 1 and 2×3 2 \times 3 tables filled with all 1's.

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