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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.

MSC:
 05A15 Exact enumeration problems, generating functions 00A08 Recreational mathematics
OEIS
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References:
 [1] J. Buhler, D. Eisenbud, R. Graham, and C. Wright, Juggling drops and descents, Amer. Math. Monthly 101(1994), 507–519. · Zbl 0814.05002 [2] S. Butler, F. Chung, J. Cummings, and R. Graham, Juggling card sequences, J. Comb. 8(2017), 507–539. · Zbl 1370.05011 [3] S. Butler and R. Graham, Enumerating (multiplex) juggling sequences, Ann. Comb. 13 (2010), 413–424. · Zbl 1231.05009 [4] R. Ehrenborg and M. Readdy, Juggling applications to q-analogues, Discrete Math. 157 (1996), 107–125. · Zbl 0859.05010 [5] B. Polster, The Mathematics of Juggling, Springer, 2000. · Zbl 1116.00004 [6] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences,http://oeis.org. · Zbl 1044.11108 [7] J. Stadler, Juggling and vector compositions, Discrete Math. 258 (2002), 179–191. 20 · Zbl 1009.05011 [8] R. Stanley, Enumerative Combinatorics, Volume I, second edition, Cambridge, 2012.
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