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We investigate the coefficients generated by expressing the falling factorial$(xy)_k$ as a linear combination of falling factorial products $(x)_l (y)_m$for $l,m =1,...,k$. Algebraic and combinatoric properties of these coefficientsare discussed, including recurrence relations, closed-form formulae, relationswith Stirling numbers, and a combinatorial characterization in terms ofconjoint ranking tables.

Falling Factorials, Generating Functions, and Conjoint Ranking Tables