# Juggling Counts

### A collection of resources for the mathematics of juggling

## Videos:

Math and Juggling (Steve Butler; Celebration of Mind 2020) [includes juggling and "string theory"]

The mathematics of juggling (George Hart; Quanta Magazine) [video with braiding hair by juggling]

Math encounters - five balls, two hands: the patterns of juggling (Colin Wright; MoMath)

The unexpected maths of juggling (Colin Wright; CPN)

Mathematical soul of juggling (Burkhard Polster; Mathologer)

The beauty and mathematics of juggling (Alexander Leymann; TED talk)

The mathematics of jugglers (Allen Knutson; math lecture in Italy) [end includes live example of baby juggling]

Mathematics of juggling (Richard Ehrenborg; math lecture)

An introduction to the mathematics of juggling (Steve Butler; Turfe lecture)

Multiplex juggling sequences and Kostant's partition function (Pamela Harris, MAA Distinguished Lecture Series)

The mathematics of juggling patterns (Bekah Smith)

Mathematics of Juggling (Princeton juggling)

Making juggling mathematical (Erik Tou)

## Audio:

Catching and releasing (Ron Graham; AMS Math Moment)

## Books:

### Juggling counts

Published (tentatively) in 2023 by Princeton University Press.

Authors: Joe Buhler, Steve Butler, Ron Graham, Anthony Mays

Published in 2012 by Princeton University Press

Authors: Persi Diaconis and Ron Graham

Published in 2002 by Springer

Author: Burkhard Polster

## Papers:

### Overviews

Juggling mathematics and magic (Ron Graham), Notices of the International Congress of Chinese Mathematicians 1 (2013), 7-9.

Fountains, Showers, and Cascades - Juggling's quintessential combinations of algebra and acrobatics (Ron Graham and Joe Buhler), The Sciences (1984), 44-51.

The mathematics of juggling (Burkhard Polster)

Do the math!: juggling with numbers (Erik Tou), Math Horizons 21 (2014), 5-7.

Juggling performers + Math (Gregory Warrington), Math Horizons 15 (2008), 18-21.

Combinatorial aspects of juggling (Anthony Mays)

### Siteswaps / rook placements

Juggling drops and descents (Joe Buhler, David Eisenbud, Ron Graham, Colin Wright), American Mathematical Monthly 101 (1994), 507-519.

Juggling patterns, passings, and posets (Joe Buhler and Ron Graham), Mathematical Adventures for Students and Amateurs, MAA, 2004, 99-116.

A generalization of Eulerian numbers via rook placements (Esther Banaian, Steve Butler, Christopher Cox, Jeffrey Davis, Jacob Landgraf, and Scarlitte Ponce), Involve 10 (2017), 691-705.

Circulant matrices and mathematical juggling (Richard Brualdi and Michael Schroeder), Art of Discrete and Applied Mathematics 1 (2018), 17pp.

### State diagrams / primitive or prime juggling

Enumerating (multiplex) juggling sequences (Steve Butler and Ron Graham), Annals of Combinatorics 13 (2010), 413-424.

Primitive juggling sequences (Fan Chung and Ron Graham), American Mathematical Monthly 115 (2008), 185-194.

Counting prime juggling patterns (Esther Banaian, Steve Butler, Christopher Cox, Jeffrey Davis, Jacob Landgraf, and Scarlitte Ponce), Graphs and Combinatorics 32 (2016), 1675-1688.

Kostant's partition function and magic multiplex juggling sequences (Carolina Benedetti, Christopher R. H. Hanusa, Pamela E. Harris, Alejandro H. Morales, Anthony Simpson), Annals of Combinatorics 24 (2020), 439-473.

Asymptotic counting theorems for primitive juggling patterns (Erik Tou), International Journal of Number Theory 15 (2019), 1037-1050.

Juggling with pattern matching (Jean Cardinal, Steve Kremer, Stefan Langerman), Theory of Computing System 39 (2006), 425-437.

Toss and spin juggling state graphs (Harri Varpanen)

### Juggling cards

Juggling and applications to q-analogues (Richard Ehrenborg and Margaret Readdy), Discrete Mathematics 157 (1996), 107-125.

Juggling card sequences (Steve Butler, Fan Chung, Jay Cummings, Ron Graham), Journal of Combinatorics 8 (2017), 507-539.

Enumerating multiplex juggling patterns (Steve Butler, Jeongyoon Choi, Kimyung Kim, and Kyuhyeok Seo), Journal of Integer Sequences 22 (2019), 21 pp.

Determinants involving q-Stirling numbers (Richard Ehrenborg), Advances in Applied Mathematics 31 (2003), 630-642.

Juggling and vector compositions (Jonathan Stadler), Discrete Mathematics 258 (2002), 179-191.

### Universal juggling patterns

Universal juggling cycles (Fan Chung and Ron Graham), Combinatorial number theory, de Gruyter, 2006, 427-443.

Universal and overlap cycles for posets, words, and juggling patterns (Adam King, Amanda Laubmeier, Kai Orans, Anant Godbole), Graphs and Combinatorics 32 (2016), 1013-1025.

Overlap cycles for permutations: necessary and sufficient conditions (Victoria Horan), Integers 14 (2014), 12 pp.

### Random juggling

Juggling probabilities (Gregory Warrington), American Mathematical Monthly 112 (2005), 105-118.

Reverse juggling processes (Arvind Ayyer and Svante Linusson), Random Structures and Algorithms 55 (2019), 56-72.

Randomly juggling backwards (Allen Knutson), Connections in discrete mathematics, Cambridge University Press, 2018, 305-320.

Bumping sequences and multispecies juggling (Arvind Ayyer, Jérémie Bouttier, Sylvie Corteel, Svante Linusson, François Nunzi), Advances in Applied Mathematics 98 (2018), 100-126.

Geometric juggling with q-analogues (Alexander Engström, Lasse Leskelä, Harri Varpanen), Discrete Mathematics 338 (2015), 1067-1074.

Multivariate juggling probabilities (Arvind Ayyer, Jérémie Bouttier, Sylvie Corteel, François Nunzi), Electronic Journal of Probability 20 (2015), 29 pp.

Positroid varieties: juggling and geometry (Allen Knutson, Thomas Lam, David Speyer) ,Compositio Mathematica 149 (2013), 1710-1752.

Juggler's exclusion process (Lasse Leskelä and Harri Varpanen), Journal of Applied Probability 49 (2012), 266-279.

### Passing

Passing drops and descents (Cailyn Bass and Steve Butler)

### Miscellaneous

Descent polynomials for k bubble-sortable permutations of type B (Matthew Hyatt), European Journal of Combinatorics 34 (2013), 1177-1191.

A zeta function for juggling sequences (Carsten Elsner, Dominic Klyve, Erik Tou), Journal of Combinatorial Number Theory 4 (2012), 53-65.

Juggling braids and links (Satyan Devadoss and John Mugno), Mathematical Intelligencer 29 (2007), 15-22.

Braids and juggling patterns (Matthew Macauley)

Linear recurrences indexed by Z (Greg Muller)

Möbius Polynomials (Will Murray), Mathematics Magazine 85 (2012), 376-383.

The art of juggling with two balls (Steve Butler)

The mathematical art of juggling (Mike Naylor), Bridges 2011 proceedings, 33-40.