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Math 491, Problem Set \#10 \\
(due 10/21/03)
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\begin{enumerate}
\item
\begin{itemize}
\item[(a)]
Use transfer matrices to find the generating function for the number of
domino-tilings of the 4-by-$n$ grid-graph.
\item[(b)]
Use transfer matrices to find the generating function for the number of
irreducible domino-tilings of the 4-by-$n$ grid-graph. (Here an irreducible
tiling is one that cannot be decomposed into tilings of two smaller rectangles
of height 4.)
\item[(c)]
Check your work by deriving your answer to (a) from your answer to (b)
and vice versa.
\end{itemize}
\item
\begin{itemize}
\item[(a)]
Find a generating function for the number of spanning trees
of a 2-by-$n$ grid-graph.
\item[(b)]
These numbers turned up in a homework problem you did
earlier in the course. Which one was it?
\item[(c)]
Is this a coincidence, or is there a connection between
the two problems?
That is, can you find a bijection between the two sorts
of combinatorial objects?
\end{itemize}
\end{enumerate}
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