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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

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 .

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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