Counting prime juggling patterns

Abstract

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 $\gamma =\mathrm{1.32963879259...}$. For larger $b$ we show there are at least $b^{n-1}$ prime cycles of length $n$.