Open Access
Universal Juggling Cycles
Fan Chung,Ron Graham +1 more
- 01 Jan 2006
TL;DR: This note combines the ideas 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 .
read more
Abstract: 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.
read more
Chat with Paper
AI Agents for this Paper
Find similar papers on Google Scholar, PubMed and Arxiv
Write a critical review of this paper
Analyze citations of this paper to find unaddressed research gaps
Citations
Positroid varieties: juggling and geometry
TL;DR: In this paper, it was shown that the intersection of only the cyclic shifts of one Bruhat decomposition has many of the good properties of the Bruhat and Richardson decompositions.
Enumerating (Multiplex) Juggling Sequences
Steve Butler,Ron Graham +1 more
TL;DR: In this paper, the problem of enumerating periodic σ-juggling sequences of length n for multiplex juggling, where σ is the initial state (or landing schedule) of the balls, was studied.
•Journal Article
Mathematical Adventures for Students and Amateurs
11
Multivariate juggling probabilities
Arvind Ayyer,Jérémie Bouttier,Sylvie Corteel,François Nunzi +3 more
- 16 Jun 2014
TL;DR: In this paper, the authors considered refined versions of Markov chains related to juggling and showed that the stationary distribution can be obtained in finite time in one case, where the normalization factor is closed-form.
10
References
Introduction to graph theory
Abstract: In graph theory, the term graph refers to a set of vertices and a set of edges. A vertex can be used to represent any object. Graphs may contain undirected or directed edges. An undirected edge is a set of two vertices. A directed edge is an ordered pair of two vertices where the edge goes from the first vertex to the second vertex. Graphs that contain directed edges are called directed graphs or digraphs.
5K
An Introduction To The Theory Of Numbers
G. H. Hardy,Ernest M. Wright +1 more
- 31 Jul 2008
TL;DR: An Introduction to the Theory of Numbers is a classic text in elementary number theory covering key milestones and developments in the field. It is highly suitable for undergraduates and number theorists alike.
2.3K
Universal cycles for combinatorial structures
TL;DR: This paper explores generalizations of de Bruijn cycles for a variety of families of combinatorial structures, including permutations, partitions and subsets of a finite set.
196
•Book
The Mathematics of Juggling
Burkard Polster
- 01 Jan 2003
TL;DR: The Mathematics of Juggling as discussed by the authors is a collection of mostly self-contained mathematical essays that introduce the reader to many elegant results and techniques from a wide range of mathematical disciplines such as combinatorics, graph theory, knot theory, mechanics, differential equations, control theory, and robotics.
49