7/30/2019 Bibliografia Su Math Juggling

1/2

A short publication history of juggling math

Harri Varpanen

April 17, 2012

The scientific analysis of juggling patterns started around 1980, a pio-neering work being Shannon [1]. A number notation for juggling patternswas first published by Magnusson and Tiemann [2] in 1989. Graham etal. have published several papers on the combinatorics of periodic jugglingpatterns with a fixed period and a fixed number of balls [3], [4], [5], [6].Ehrenborg and Readdy [7] pointed out a connection between affine Weylgroups and periodic juggling patterns; see also Stadler [8] and Ehrenborg[9]. Knutson, Lam and Speyer [10] (see also Snider [11]) have recently con-tinued along a related line, indexing positroid varieties in the Grassmannianby periodic juggling patterns. Some connections between periodic patternsand braid groups have been studied by Devadoss and Mugno [12], and Pol-ster [13] has written a monograph about various (periodic) juggling mathideas.

Warrington [14] deviated from the realm of periodic patterns and cal-culated the steady-state distribution for patterns having a fixed number ofballs and a bounded, uniformly distributed random throw height. Leskelaand Varpanen [15] continued in the unbounded setting, proving that a largeclass of patterns have a unique steady-state distribution. They also calcu-lated the steady-state distribution for unbounded patterns with a shifted-geometric throw distribution.

References

[1] Claude Shannon. Scientific Aspects of Juggling. Manuscript from ca.

1980. Published in Claude Elwood Shannon, Collected Papers (Wiley1993), 850864.

[2] Bengt Magnusson and Bruce Tiemann. The Physics of Juggling. Phys.Teach. 27, 584588 (1989)

[3] Joe Buhler, David Eisenbud, Ron Graham and Colin Wright. Jugglingdrops and descents. Organic mathematics (Burnaby, BC, 1995), 133154, CMS Conf. Proc., 20, Amer. Math. Soc., Providence, RI, 1997.

[4] Fan Chung and Ron Graham. Primitive juggling sequences. Amer.Math. Monthly 115 (2008), no. 3, 185194

1

7/30/2019 Bibliografia Su Math Juggling

2/2

[5] Steve Butler and Ron Graham. Enumerating (multiplex) juggling se-quences. Ann. Comb. 13 (2010), no. 4, 413424.

[6] Fan Chung, Anders Claesson, Mark Dukes, and Ron Graham. Descentpolynomials for permutations with bounded drop size. European J.Combin. 31 (2010), no. 7, 18531867

[7] Richard Ehrenborg and Margaret Readdy. Juggling and applicationsto q-analogues. Discrete Math. 157 (1996), no. 1-3, 107125.

[8] Jonathan D. Stadler. Juggling and vector compositions. Discrete Math.258 (2002), no. 1-3, 179191.

[9] Richard Ehrenborg. Determinants involving q-Stirling numbers. Adv.

Appl. Math. 31:630642, 2003.

[10] Allen Knutson, Thomas Lam, and David E Speyer. Positroid varietiesI: juggling and geometry. arXiv:0903.3694.

[11] Michelle Snider. Affine Patches on Positroid Varieties and Affine PipeDreams (Ph.D. Thesis). arXiv:1011.3705.

[12] Satyan Devadoss and John Mugno. Juggling braids and links. Math.Intelligencer 29 (2007), no. 3, 1522.

[13] Burkard Polster. The Mathematics of Juggling. Springer Verlag, 2003.

[14] Gregory S. Warrington. Juggling probabilities. Amer. Math. Monthly112(2):105118, 2005.

[15] Lasse Leskela and Harri Varpanen. Jugglers exclusion process. J. Appl.Probab. 49(1), 2012, to appear.

2