connection between simple permutations and inflations is furnished by the following result.

For every permutation $\pi$ there is a unique simple permutation $\sigma$ such that $\pi=\sigma[\alpha_1,\alpha_2,\ldots,\alpha_m]$. Furthermore, except when $\sigma=12$ or $\sigma=21$, the intervals of $\pi$ that correspond to $\alpha_1,\alpha_2,\ldots,\alpha_m$ are uniquely determined. In the case that $\sigma=12$ (respectively $\sigma=21$), the intervals are unique so long as we require the first of the two intervals to be sum (respectively skew) indecomposable.

One of the first general enumeration results is the following from [@albert:simple-permutat:]:

\[thm-fin-simples-alg\] If the class ${\mathcal{C}}$ contains only finitely many simple permutations, then ${\mathcal{C}}$ has an algebraic generating function.

This theorem has since been generalised in two different directions. Brignall, Huczynska, and Vatter [@brignall:simple-permutat:] introduced the notion of “query-complete sets of properties” to show that if a class satisfies the hypotheses of Theorem \[thm-fin-simples-alg\], then such subsets as the even permutations or the involutions in ${\mathcal{C}}$ have algebraic generating functions. More relevant to our investigation, [@albert:inflations-of-g:] significantly weakened the hypotheses of Theorem \[thm-fin-simples-alg\], showing that its conclusion holds even when ${\mathcal{C}}$ contains infinitely many simple permutations, so long as these simple permutations lie in a geometric grid class, a notion introduced in Section \[sec-geom-grid-review\]. Before this, we consider 