Falling Factorials, Generating Functions, and Conjoint Ranking Tables
Abstract
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 coefficients are discussed, including recurrence relations, closed-form formulae, relations with Stirling numbers, and a combinatorial characterization in terms of conjoint ranking tables.
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