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 longstanding 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 twoball prime juggling patterns of length $n$ is $(\gammao(1))2^n$ where $\gamma=1.32963879259...$. For larger $b$ we show there are at least $b^{n1}$ prime cycles of length $n$.
 Publication:

arXiv eprints
 Pub Date:
 August 2015
 arXiv:
 arXiv:1508.05296
 Bibcode:
 2015arXiv150805296B
 Keywords:

 Mathematics  Combinatorics