&=&0.
\end{array}$$

The first several terms of this sequence are $$1,2,6,22,88,367,1568,6810,29943,132958,595227, 2683373,12170778,55499358,$$ sequence [A165538]{} in the [OEIS [@sloane:the-on-line-enc:]]{}. Though the form of the equation for $f$ is complicated, the close link with the Catalan numbers which can be seen in the previous development is enough to ensure that the radius of convergence is quite simple, exactly $1/5$, and in particular $f_n^{1/n} \to 5$.

Example \#3: Avoiding $4231$ and $3124$ {#sec-4231-3124}
=======================================

Before our final example we review a well-studied class. A permutation is [*layered*]{} if it is the direct sum of decreasing permutations (these decreasing permutations are called the [*layers*]{}). The class of [*layered permutations*]{} has the basis $\{312, 231\}$. To restrict the number of layers, we merely need to add an additional restriction, of the form $12\cdots k$.

\[prop-4231-3124-simples\] The simple permutations of ${\operatorname{Av}}(4231, 3124)$ and ${\operatorname{Geom}}{\mbox{\begin{footnotesize}$\left(\begin{array}{rrr}0&1&-1\\1&-1&0\end{array}\right)$\end{footnotesize}}}$ coincide.

First, it is straightforward to observe that $${\operatorname{Geom}}{\mbox{\begin{footnotesize}$\left(\begin{array}{rrr}0&1&-1\\1&-1&0\end{array}\right)$\end{footnotesize}}}
=
{\operatorname{Grid}}{\mbox{\begin{footnotesize}$\left(\begin{array}{rrr}0&1&-1\\1&-1&0\end{array}\right)$\end{footnotesize}}}
\subseteq{\operatorname{Av}}(4231, 3124),$$ so it suffices to prove that the simple permutations of ${\operatorname{Av}}(4231, 3124)$ are contained in this grid class.

Consider a simple permutation $\pi\in{\operatorname{Av}}(4231,3124)$ of length $n$. We analyse the entries of $\pi$ to the left of $n$ and to the right of $n$ separately, beginning with the entries on 