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.