Universal Juggling Cycles

During the past several decades, it has become popular among jugglers (and juggling mathematicians) to represent certain periodic juggling patterns by sequences of non-negative numbers. These sequences, usually called “site-swaps” in the juggling vernacular, have been studied in a variety of papers, and are known to have many interesting properties. Another idea in combinatorics that has emerged in the past 10 − 15 years or so is that of a “universal” sequence, in which one represents a whole class C of sequences by one long cyclic sequence U, where each sequence in C is obtained by a shifting a “window” moving along U. In this note, we combine these ideas to construct universal juggling cycles. How long can these universal juggling cycles be? How many of these universal cycles are needed to represent all the juggling patterns? We answer these questions and raise several new ones.