Enumerating Prime Patterns in Juggling Variations
Fuente:
arXiv
Saved in:
| Main Authors: | , , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910057702096896 |
|---|---|
| author | Butler, Steve Choi, Vera Jeffries, Joel McCambridge, Nina Morgenstern, Asia Mateo, Samuel Orellana |
| author_facet | Butler, Steve Choi, Vera Jeffries, Joel McCambridge, Nina Morgenstern, Asia Mateo, Samuel Orellana |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17284 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Enumerating Prime Patterns in Juggling Variations Butler, Steve Choi, Vera Jeffries, Joel McCambridge, Nina Morgenstern, Asia Mateo, Samuel Orellana Combinatorics 05A15, 05A05 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. |
| title | Enumerating Prime Patterns in Juggling Variations |
| topic | Combinatorics 05A15, 05A05 |
| url | https://arxiv.org/abs/2603.17284 |