Joe Buhler, David Eisenbud, Ron Graham, and Colin Wright
As circus and vaudeville performers have known for a long time, juggling is fun. In the last twenty years or so this has led to a surge in the number of amateur jugglers. It has been observed that scientists, and especially mathematicians and computer scientists, are disproportionately represented in the juggling community. It is difficult to explain this connection in any straightforward way, but music has long been known to be popular among scientists; juggling, like music, combines abstract patterns and mind-body coordination in a pleasing way. In any event, the association between mathematics and juggling may not be as recent as it appears, since it is believed that the tenth century mathematician Abu Sahl started out juggling glass bottles in the Bagdad marketplace (, p. 79).
In the last fifteen years there has been a corresponding increase in the application of mathematical and scientific ideas to juggling (, , , , , ), including, for instance, the construction of a juggling robot (). In this article we discuss some of the mathematics that arises out of a recent juggling idea, sometimes called ``site swaps.'' It is curious that these idealized juggling patterns lead to interesting mathematical questions, but are also of considerable interest to ``practical'' jugglers. The basic idea seems to have been discovered independently by a number of people; we know of three groups or individuals that developed the idea around 1985: Bengt Magnusson and Bruce Tiemann (, ), Paul Klimek in Santa Cruz, and one of us (C. W.) in conjunction with other members of the Cambridge University Juggling Association. A precursor of the idea can be found in .
Although our interests here are almost entirely mathematical, the reader interested in actual juggling or its history might start by looking at  and ; a leisurely discussion of site swaps, aimed at jugglers, can be found in .
In the first section we describe the basic ideas, and in the second section we prove the basic combinatorial result that counts the number of site swaps with a given period and a given number of balls. This theorem has a non-obvious generalization to arbitrary posets (). Special cases of that result can be interpreted in terms of an interesting generalization of site swaps; we find it delightful that a question arising from juggling leads to new mathematics which in turn may say something about patterns that jugglers might want to consider.