### 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

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