zbMATH — the first resource for mathematics

Enumerating (Multiplex) juggling sequences. (English) Zbl 1231.05009
Summary: We consider the problem of enumerating periodic \(\sigma\)-juggling sequences of length \(n\) for multiplex juggling, where \(\sigma\) is the initial state (or landing schedule) of the balls. We first show that this problem is equivalent to choosing 1’s in a specified matrix to guarantee certain column and row sums, and then using this matrix, derive a recursion. This work is a generalization of earlier work of Chung and Graham.

05A15 Exact enumeration problems, generating functions
00A08 Recreational mathematics
Full Text: DOI arXiv
[1] Buhler J., Eisenbud D., Graham R., Wright C. (1994) Juggling drops and descents. Amer. Math. Monthly 101: 507–519 · Zbl 0814.05002
[2] Buhler J., Graham R. (1994) A note on the binomial drop polynomial of a poset. J. Combin. Theory Ser. A 66: 321–326 · Zbl 0797.06002
[3] Buhler, J., Graham, R.: Juggling patterns, passing, and posets. In: Hayes, D.F., Shubin, T. (eds.) Mathematical Adventures for Students and Amateurs, pp. 99–116. The Mathematical Association of America, Washington (2004)
[4] Chung, F., Graham, R.: Universal juggling cycles. Integers 7(2), #A8 (2007) · Zbl 1123.05005
[5] Chung F., Graham R. (2008) Primitive Juggling Sequences. Amer. Math.Monthly 115(3): 185–194 · Zbl 1170.05006
[6] Ehrenborg R., Readdy M. (1996) Juggling and applications to q-analogues. Discrete Math. 157: 107–125 · Zbl 0859.05010
[7] Gessel I.M., Stanley R.P. (1995) Algebraic enumerations. In: Graham R.L., Grötschel M., Lovész L. (eds) Handbook of Combinatorics Vol II. Elsevier, Amsterdam, pp 1021–1061 · Zbl 0853.05002
[8] Graham R.L., Knuth D.E., Patashnik O. (1994) Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley, Reading, MA · Zbl 0836.00001
[9] Juggling Information Service. http://www.juggling.org/
[10] Polster B. (2000) The Mathematics of Juggling. Springer, New York · Zbl 1116.00004
[11] Sloane, N.: The On-Line Encyclopedia of Integer Sequences. Avaible at: http://www.research.att.com/\(\sim\)njas/sequences/ · Zbl 1274.11001
[12] Stadler J.D. (2002) Juggling and vector compositions. Discrete Math. 258: 179–191 · Zbl 1009.05011
[13] Stadler, J.D.: personal communication
[14] Warrington G.S. (2005) Juggling probabilities. Amer. Math. Monthly 112(2): 105–118 · Zbl 1078.60011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.