Counting prime juggling patterns

TL;DR

This paper develops formulas for counting prime juggling patterns, linking them to partitions and providing asymptotic estimates, especially for the case of two balls.

Contribution

It introduces a new method to count prime juggling patterns, connecting cycle counts to partitions and deriving asymptotic formulas for two-ball patterns.

Findings

Number of two-ball prime patterns of length n is approximately 1.33... times 2^n.

For larger numbers of balls, there are at least b^{n-1} prime cycles.

Established a connection between prime juggling patterns and partitions of n.

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