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Enumerating Prime Patterns in Juggling Variations

Steve Butler, Vera Choi, Joel Jeffries, Nina McCambridge, Asia Morgenstern, Samuel Orellana Mateo

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.

Enumerating Prime Patterns in Juggling Variations

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 -ball prime patterns with period . 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, , 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.
Paper Structure (37 sections, 32 theorems, 130 equations, 6 figures, 4 tables)

This paper contains 37 sections, 32 theorems, 130 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. State Graph
  3. Work on prime patterns and formalization
  4. Variations of Juggling
  5. Infinite State Graph
  6. Cards Notation
  7. Our work
  8. Asymptotics for c_t(n)
  9. Preliminaries
  10. General Asymptotic Theorem
  11. Multiplex 2-Ball
  12. Preliminaries
  13. Set Notation
  14. Main Theorem
  15. Asymptotics
  16. ...and 22 more sections

Key Result

Lemma 1.1

Let $\mathcal{X}_{t,n}$ the set of all $X_{t,n} = (S_1, \dots, S_t)$ (up to cyclic order), where and $S_i \cap S_j = \varnothing$ for all $1 \leq i < j \leq t$. This is possibly empty if $t$ is not large enough. Then, there is a bijection between the set of prime patterns $\rho$ of length $n$ that contain $t$$C_2$ cards in their construction, and elements in $\mathcal{X}_{t,n}$.

Figures (6)

  • Figure 1: A subgraph of the two-ball state graph. The edge labels correspond to throw heights.
  • Figure 2: Cards for $b = 2$
  • Figure 3: A portion of the state diagram when $b=3$ and $k=2$.
  • Figure 4: An example of a 2-ball multiplex pattern.
  • Figure 5: New cards used in strict multiplex patterns
  • ...and 1 more figures

Theorems & Definitions (70)

  • Lemma 1.1
  • Definition 1.1
  • Proposition 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1
  • Definition 2.4
  • Theorem 2.2
  • Theorem 2.3: General Asymptotic Theorem
  • ...and 60 more