Theorem 2

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


Suppose that f is a juggling pattern and . Then so there is an integer-valued function such and


and the stated condition is satisfied.

Conversely, suppose that

is such that is a permutation of . If we define for all integers t by extending the sequence periodically and then define then f is the desired juggling pattern. To see that f is injective note that if then since is injective modulo n. Then . From it follows that t = u and f is injective as claimed. To show that f is surjective, suppose that . Since is a permutation of we can find a a t such that . By adding a suitable multiple of n we can find a such that . This finishes the proof of the fact that any sequence satisfying the stated condition comes from a juggling pattern.