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Math 192r, Problem Set \#16 \\
(due 11/20/01)
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\begin{enumerate}
\item
Use Dodgson condensation to prove the
Vandermonde determinant formula
$$\det(M)=\prod_{1 \leq i < j \leq n} (x_j-x_i)$$
where $M$ is the $n$-by-$n$ matrix
whose $i,j$th entry (for $1 \leq i,j \leq n$)
is $x_j^{i-1}$.
\item
Using Dodgson condensation, Lindstrom's lemma,
and the bijection between tilings and routings discussed in class,
prove that for all $a,b,c \geq 0$,
the number of ways to tile an $a,b,c,a,b,c$
semiregular hexagon with unit rhombuses is equal to
$$\frac{H(a+b+c)H(a)H(b)H(c)}{H(a+b)H(a+c)H(b+c)}$$
where
$H(0)=H(1)=1$ and
$H(n)=1!2!3! \cdots (n-1)!$ for $n > 1$.
\end{enumerate}
\noindent
For both of these problems, you should
use only the properties of the determinant
that were discussed in lecture
(or that you prove yourself).
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