$\sum |{\mathcal{C}}_n|x^n$ or simply an asymptotic result about the behaviour of $|{\mathcal{C}}_n|$ as $n\rightarrow\infty$.

Recently, Albert, Atkinson, Bouvel, Ruškuc, and Vatter [@albert:geometric-grid-:] have developed the theory of geometric grid classes, and Albert, Ruškuc, and Vatter [@albert:inflations-of-g:] have continued this exploration by investigating the theory of inflations of such classes. Our aim in this paper is to:

-   demonstrate the effectiveness of this approach, and

-   illustrate how one might implement these techniques in a “real world” setting, bypassing what would otherwise be thorny theoretical issues.

It should be noted that this presentation is historically backward; the results of this paper preceded and inspired the two more theoretical papers cited above.

In this work, our examples are exclusively classes with two basis elements of length four, which we call *2 $\times$ 4 classes*. It must be admitted that the attention paid to 2 $\times$ 4 classes is not entirely in proportion to their intrinsic importance. Nevertheless, these classes represent a significant dataset which seems to contain some difficult enumerative problems. Thus they pose a good challenge for new approaches to the enumeration of restricted permutations.

There are $56$ essentially different (i.e. inequivalent under symmetries) 2 $\times$ 4 classes. Some of these 