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Combinatorics of Falling Factorials


ff_subarrayQuestion: What do the Fourier uncertainty principle, Stirling numbers, and the EE Quals have in common?

Answer: Falling factorials.

We investigate the coefficients c_{l,m}^{(k)} generated by expressing the falling factorial (xy)^{falling k} as a linear combination of falling factorial products (xy)^{falling k) = \sum_{1 \leq l,m \leq k} c_{l,m}^{(k)} x^{falling l} y^{falling m}. Algebraic and combinatorial properties of these coefficients are discussed, in relation to Stirling numbers, binomial coefficients, and generating functions for conjoint ranking tables.