Journal of Integer Sequences, Vol. 22 (2019), Article 19.1.7

Enumerating Multiplex Juggling Patterns

Steve Butler
Department of Mathematics
Iowa State University
Ames, IA 50011

Jeongyoon Choi, Kimyung Kim, and Kyuhyeok Seo
Gyeonggi Science High School for the Gifted
Gyeonggi Province
Republic of Korea


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 κ, and include data for κ = 2, 3, and establish some combinatorial properties of the cards.

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(Concerned with sequences A000041 A000712 A003480 A090764 A178841.)

Received March 13 2017; revised versions received March 2 2018; January 30 2019; January 31 2019. Published in Journal of Integer Sequences, February 2 2019.

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