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Math 192r, Problem Set \#11 \\
(due 10/25/01)
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\begin{enumerate}
\item
An ``augmented Aztec diamond of order $n$''
is a subset of the square grid
formed by symmetrically stacking rectangles of height 1
and respective widths $2,4,6,\dots,2n-2,2n,2n,2n,2n-2,\dots,6,4,2$
(note that there are 3 rectangles of width $2n$ being stacked).
Here, for instance, is an augmented Aztec diamond of order 3:
$$
\pspicture(-1,-1)(7,8)
\pspolygon(0,2)(0,5)(1,5)(1,6)(2,6)(2,7)(4,7)(4,6)(5,6)(5,5)(6,5)(6,2)(5,2)(5,1)(4,1)(4,0)(2,0)(2,1)(1,1)(1,2)(0,2)
\endpspicture
$$
Express the generating function $\sum_{n \geq 0} a_n t^n$ 
as an algebraic function of $t$,
where $a_n$ denotes the number of domino-tilings
of the augmented Aztec diamond of order $n$,
and we put $a_0=1$.
(Note that the augmented Aztec diamond of order 1
is just a rectangle of height 3 and width 2.)
\item
Use the exchange principle for 2-routings to find
a compact formula for the number of lozenge-tilings
of the semiregular hexagon with side-lengths $a,b,2,a,b,2$.
\end{enumerate}
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