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Enumerating multiplex juggling patterns. (English) Zbl 1407.05011
Summary: A classic problem in the mathematics of juggling is to give a basic enumeration of the number of juggling patterns. This has been solved in the case when at most one ball is caught/thrown at a time, with the simplest proof being due to Ehrenborg and Readdy by the use of cards.
We introduce a new set of cards that can be used to count multiplex juggling patterns (when multiple balls can be caught/thrown at a time). This set of cards models the correct behavior and avoids the problems of ambiguity; on the other hand the cards are no longer independent. By use of the transfer matrix method combined with the cards we enumerate multiplex juggling patterns with exactly \(b\) balls and hand capacity \(\kappa\), and include data for \(\kappa=2,3\), and establish some combinatorial properties of the cards.

05A15 Exact enumeration problems, generating functions
00A08 Recreational mathematics
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