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\lecture{2004/05/07}{Dan Ports}{2004/05/17}{drkp@mit.edu}
\lecturetitle{Site-Swap Notation}{Marc Strauss, MIT Undergraduate
Journal of Mathematics}{Marc Strauss}{Dan Ports}

Site-swap notation is a means for compactly expressing juggling
patterns. Without site-swap notation, we might choose to express a
juggling pattern as a graph, with a set of nodes corresponding to each
hand's state at each step in the sequence. Each node has an incoming
edge representing the ball that is received by that hand at that point
in the sequence, and an outgoing edge that indicates where the ball is
thrown to. While this captures the full set of information about a
juggling sequence, it is rather cumbersome, as the example in
Figure~\ref{fig:graph} demonstrates. Site-swap notation places only a
few additional restrictions on the juggling pattern, and allows the
same information to be expressed in a short numeric form.

\begin{figure}[htbp]
  \centering
  \vspace{2em}
\begin{verbatim}
    ...   L     L     L     L     L     L     L     L     L     L     L             
      \  @@@@  __/\  __/\  @@@@  __/\  __/\  @@@@  __/\  __/\  @@@@             
     @@\@  __@/  __\/  @@\@  __@/  __\/  @@\@  __@/  __\/  @@\@  __@/      51   
       _\_/  _@_/  @\@@  _\_/  _@_/  @\@@  _\_/  _@_/  @\@@  _\_/               
    __/  \__/  @@@@  \__/  \__/  @@@@  \__/  \__/  @@@@  \__/  \__/             
...R     R     R     R     R     R     R     R     R     R     R                

\end{verbatim}
  \caption{Example of a graphical representation of a juggling pattern, with one ball singled out. Source: Knutson \cite{knutson}}
  \label{fig:graph}
\end{figure}

We divide a juggling pattern into a series of discrete events called
``throws'' --- these correspond to a ball being caught in one hand and
then immediately being thrown into the air. A juggling pattern is a
sequence of throws. We are interested, at least initially, in periodic
sequences, though of course it is possible in some cases to interject
``tricks'' consisting of a different pattern into the middle of a
juggling pattern (we will return to this notion later). Site-swap
notation numbers each throw according to the number of other throws
that take place while that ball is in the air. In order to make the
notation even more compact, we apply the two restrictions (which
Knutson refers to as ``vanilla site-swap notation'' \cite{knutson})
that only one ball is thrown at a a time, and they are always thrown
in sequence with each hand alternating. These restrictions do
eliminate a number of interesting juggling patterns, but they still
allow for many others, and they can now be expressed as a simple
stream of numbers. We note that the sequence repeats, and write only
the numbers corresponding to one period.

For generality, we also assign the number zero to represent that a
hand is empty while the other hand does two in a row, which we might
call a ``virtual throw'' since it is a throw position at which no ball
is actually thrown. Knutson says ``It is simpler to think of a 0-throw
as one where there is actually a ball borrowed from the Astral Plane
only long enough to throw and catch at the exact same time - then at
each time, we're always throwing, and catching, exactly one ball. This
is the main reason why it is mathematically nice to call these throws
0s.'' \cite{knutson}. I can't argue with that.

With site-swap notation now defined, we turn to looking at some
interesting properties of it:

\begin{itemize}
\item In order for an arbitrary list of numbers to be a valid
  site-swap notation for a juggling pattern, two conditions must hold.
  The two are equivalent, so only one needs to be checked.
  \begin{enumerate}
  \item At every step, at most one ball must be caught --- no two
    throws land in the same place.
  \item At every step, one ball must be caught (modulo the ``0 case'')
    in order to be thrown.
  \end{enumerate}
  
\item A throw throws the ball back to the same hand if the number
  corresponding to it is even, and to the opposite hand if the number
  corresponding to it is odd.

\item If a pattern has an even number of numbers in it, the throws
  always take place from the same hands. Otherwise, the throws
  alternate hands, and the pattern is \emph{symmetric} since the same
  throws are performed by each hand.
  
\item To find out the number of balls required by a given pattern,
  take the average of the numbers in the site-swap notation. This will
  be an integer if it is a valid pattern (though this condition is not
  sufficient --- it is easy to find an example where the average is an
  integer but the pattern is not valid according to the two conditions
  above), and this integer is the number of balls required.

\end{itemize}

Of course, knowing about the notation does not make it possible to
juggle a given pattern --- this is a much harder task, and one which I
would have no hope of accomplishing.

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