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Juggling and vector compositions. (English) Zbl 1009.05011
Author’s abstract: A considerable amount of interest has arisen pertaining to the mathematics of juggling. In this paper, we use techniques similar to those of R. Ehrenborg and M. Readdy [Discrete Math. 157, 107-125 (1996; Zbl 0859.05010)] to enumerable two specal classes of juggling patterns. Both classes are enumerated by a product of \(q\)-binomial coefficients; the second class is a generalization of the first. Using the first class of patterns, we give a bijective proof of an identity of J. Haglund [Composition, rook placements, and permutations of vectors (Doctoral Dissertation, University of Georgia, Athens, GA) (1993)] involving vector compositions. We define a generalized vector composition and provide a bijective proof of an identity dual to Haglund’s using these generalized vector compositions.

MSC:
05A15 Exact enumeration problems, generating functions
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