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Math 192r, Problem Set \#12 \\
(due 11/6/01)
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\begin{enumerate}
\item
We consider directed animals
on the modified square lattice
that has an extra edge joining $(i,j)$ to $(i+1,j+1)$
for all $i,j$.
A subset $S$ of the first quadrant is a directed animal on this lattice
if for every point $(i,j)$ in $S$ 
there is a path from $(0,0)$ to $(i,j)$ in $S$
via steps of the form $(+1,0)$, $(0,+1)$, $(+1,+1)$.
Let $a_n$ be the number of directed animals on this lattice
having $n$ elements,
so that $a_1=1$, $a_2=3$, $a_3=10$, etc.
Mimic the method discussed in class for the ordinary square lattice
to derive a formula for the generating function 
$\sum_{n=1}^{\infty} a_n$,
and use this to obtain a formula for $a_n$ itself
as well as a formula for $\lim_{n \rightarrow \infty} a_n^{1/n}$.
\item
\begin{itemize}
\item[(a)]
The mapping from the ring of formal power series to itself
that sends $f(x)$ to $1+x^2[f(x)]^3$
has a unique fixed point.
Conjecture a formula for the coefficients
of this formal power series.
(Hint: Try to express the ratio of the coefficients
of $x^{2n}$ and $x^{2n-2}$ as a rational function of $n$.)
\item[(b)]
There exist Laurent series
$$g(x) = x^{-1} - \frac{1}{2} - \frac{3}{8}x 
- \frac{1}{2}x^2 - \dots$$
and
$$g(-x) = - x^{-1} - \frac{1}{2} + \frac{3}{8}x 
- \frac{1}{2}x^2 + \dots$$
that are also fixed under that mapping.
Find the first dozen coefficients of $g$
and conjecture a formula for the coefficient of $x^n$.
\end{itemize}
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