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# Periodic Juggling

From now on we want to juggle periodically. A juggling pattern is perceived to be periodic by an audience when its height function is periodic in the mathematical sense.

Definition:

A period-n juggling pattern is a bijection such that for all .

If df is of period n then it might also have a period m for some divisor m of n. If n is the smallest period of df then any other period is a multiple of n; in this case we will say that f is a pattern of exact period n.

A period-n juggling pattern can be described by giving the finite sequence of non-negative integers for . Thus the pattern 51414 denotes a period-5 pattern; by Theorem 1 it is a 3-ball pattern since the ``period average'' of the height function is 3.

Which finite sequences correspond to juggling patterns? Certainly a necessary condition is that the average must be an integer. However this isn't sufficient. The sequence 354 has average 3 but does not correspond to a juggling pattern---if you try to draw an arrow diagram for a map f as above you'll find that no such map exists. This is also easy to see directly, for if and then

and such a map isn't a bijection.

### Lemma 1

If f is a period-n juggling pattern then

### [Proof]

The Lemma implies that a juggling pattern f induces a well-defined injective, and hence bijective, mapping on the integers modulo n. Let denote the set and let denote the symmetric group consisting of all permutations (bijections) of the set . Then for every period n juggling pattern f there is a well-defined permutation that is defined by the condition

### Theorem 2

A sequence of non-negative integers satisfies for some period-n juggling pattern f if and only if is a permutation of .

### [Proof]

To see if 345 corresponds to a juggling pattern we add t to the t-th term and reduce modulo 3. The result is 021, which is a permutation, so 345 is indeed a juggling pattern (in fact a somewhat difficult one that is quite amusing). On the other hand, the sequence 354 leads, by the same process, to 000 which certainly isn't a permutation of .

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