# 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)

## 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.