Enumerating Prime Patterns in Juggling Variations

TL;DR

This paper develops a comprehensive combinatorial framework for enumerating various prime juggling patterns, providing new bounds, exact counts, and asymptotic analyses for different juggling variations.

Contribution

It introduces a unified state-based approach and combinatorial tools to enumerate prime juggling patterns, including new bounds and exact counts for multiple variations.

Findings

Established a new lower bound on the number of b-ball prime patterns with period n.

Determined exact counts for 2-ball multiplex, 1-ball passing, and 2-ball colored patterns.

Analyzed asymptotic growth rates of pattern counts.

Abstract

Juggling patterns can be mathematically modeled as closed walks within directed state graphs. In this paper, we present a unified framework of unbounded juggling patterns and its variations (including multiplex, colored, and passing) primarily through the formalism of the juggling state. By extending this state-based approach and utilizing combinatorial tools such as set partitions and filled Ferrers diagrams, we find and prove a new lower bound on the number of $b$-ball prime patterns with period $n$. Further, we determine exact counts for 2-ball multiplex, 1-ball passing, and 2-ball colored juggling patterns, as well as a lower bound for 2-ball passing. We also provide an extensive analysis of the asymptotic growth rates for these pattern counts. Finally, we formalize the infinite state graph, $G_\infty$, and utilize flip-reverse involutions to establish bijections between classes of prime patterns, exploring how fixing a specific state influences the enumeration of prime walks.