Counting Prime Juggling Patterns


Juggling patterns can be described by a closed walk in a (directed) state graph, where each vertex (or state) is a landing pattern for the balls and directed edges connect states that can occur consecutively. The number of such patterns of length n is well known, but a long-standing problem is to count the number of prime juggling patterns (those juggling patterns corresponding to cycles in the state graph). For the case of \(b=2\) balls we give an expression for the number of prime juggling patterns of length n by establishing a connection with partitions of n into distinct parts. From this we show the number of two-ball prime juggling patterns of length n is \((\gamma -o(1))2^n\) where \(\gamma =1.32963879259\ldots \). For larger b we show there are at least \(b^{n-1}\) prime cycles of length n.

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The authors are grateful for many useful comments and discussions with Ron Graham on the mathematics of juggling. The research was conducted at the 2015 REU program held at Iowa State University which was supported by NSF DMS 1457443.

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Correspondence to Steve Butler.

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Banaian, E., Butler, S., Cox, C. et al. Counting Prime Juggling Patterns. Graphs and Combinatorics 32, 1675–1688 (2016).

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  • Juggling
  • State graphs
  • Enumeration
  • Asymptotics
  • Partitions