behaved.

\[thm-geom-griddable-all\] Suppose that ${\mathcal{C}}\subseteq{\operatorname{Geom}}(M)$ is a permutation class and $M$ is a partial multiplication matrix with cell alphabet $\Sigma$. Then the following hold:

1.  ${\mathcal{C}}$ is partially well-ordered.

2.  ${\mathcal{C}}$ is finitely based.

3.  There is a regular language $L\subseteq\Sigma^\ast$ such that the mapping ${\varphi}{\::\:}L\rightarrow{\mathcal{C}}$ is a bijection.

4.  There is a regular language $L_S$, contained in the regular language $L$ from (iii), such that the mapping ${\varphi}$ is a bijection between $L_S$ and the simple permutations in ${\mathcal{C}}$.

Note that the proof of Theorem \[thm-geom-griddable-all\] is nonconstructive, so while we use the encoding ${\varphi}$ throughout this work, we construct the regular languages we use from first principles.

Albert, Atkinson, and Brignall [@albert:the-enumeration:2143:4231; @albert:the-enumeration:3] demonstrate four concrete examples of using these techniques to enumerate $2\times 4$ classes.

The examples considered in this paper are inflations of geometric grid classes. The theoretical issues of such classes were studied by Albert, Ruškuc, and Vatter [@albert:inflations-of-g:], who proved that every subclass of $\langle{\operatorname{Geom}}(M)\rangle$ has an algebraic generating function (essentially by showing that it is in bijection with a context-free language). From this perspective, Section \[sec-4213-3142\] considers the case of a class contained in $\langle{\operatorname{Geom}}(\begin{footnotesize}\begin{array}{rr}1&1\end{array}\end{footnotesize})\rangle$, while the next two sections consider classes whose 