<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume 13 Issue 3" %>
Enumerating (Multiplex) Juggling Sequences
Steve Butler1 and Ron Graham2
1Department of Mathematics, University of California, Los Angeles, CA 90095, USA
butler@math.ucla.edu
2Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA
graham@ucsd.edu
Annals of Combinatorics 13 (4) pp.413-424 December, 2009
AMS Subject Classification: 00A08; 05A15
Abstract:
We consider the problem of enumerating periodic s-juggling sequences of length n for multiplex juggling, where s is the initial state (or landing schedule) of the balls. We first show that this roblem 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.
Keywords: juggling, multiplex, state diagram, recursions

References:

1. Buhler, J., Eisenbud, D., Graham, R., Wright, C.: Juggling drops and descents. Amer. Math. Monthly 101, 507--–519 (1994)

2. Buhler, J., Graham, R.: A note on the binomial drop polynomial of a poset. J. Combin. Theory Ser. A 66, 321--–326 (1994)

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)

5. Chung, F., Graham, R.: Primitive Juggling Sequences. Amer. Math. Monthly 115(3), 185-- 194 (2008)

6. Ehrenborg, R., Readdy, M.: Juggling and applications to q-analogues. Discrete Math. 157, 107–--125 (1996)

7. Gessel, I.M., Stanley, R.P.: Algebraic enumerations. In: Graham, R.L., Götschel, M., Loész, L. (eds.) Handbook of Combinatorics Vol. II, pp. 1021--–1061. Elsevier, Amsterdam (1995)

8. Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley, Reading, MA (1994)

9. Juggling Information Service. http://www.juggling.org/

10. Polster, B.: The Mathematics of Juggling. Springer, New York (2000)

11. Sloane, N.: The On-Line Encyclopedia of Integer Sequences. Avaible at: http://www.research.att.com/finjas/sequences/

12. Stadler, J.D.: Juggling and vector compositions. Discrete Math. 258, 179--–191 (2002)

13. Stadler, J.D.: personal communication

14. Warrington, G.S.: Juggling probabilities. Amer. Math. Monthly 112(2), 105--–118 (2005)