Counting the Number of Site Swap Juggling Patterns with Respect to Particular Ceilings

TL;DR

This paper develops formulas to count site swap juggling patterns considering specific ceiling height restrictions, addressing a complex problem in mathematical juggling pattern enumeration.

Contribution

It introduces new formulae for counting juggling patterns with ceiling constraints, extending prior work on unrestricted pattern enumeration.

Findings

Derived formulae for specific ceiling types

Extended enumeration methods to constrained patterns

Addressed open problem in juggling pattern counting

Abstract

Site swap is a mathematical notation used by jugglers to communicate, create and study complex juggling patterns. Determining the number of possible site swap juggling patterns with respect to certain limiting parameters such as number of balls etc., is a problem that has been much studied and solved by many mathematicians. However, when the patterns have a throw height restriction (ceiling) the problem becomes difficult and is in general still open. In this article we derive some formulae for computing the number of possible juggling patterns with respect to certain ceiling types.