be performed automatically using the Maple package [FinLabel]{} described in Vatter [@vatter:finitely-labele:].

-   ${\operatorname{Av}}(4213,3124)$ contains precisely two simple permutations of each length $n\ge 4$, oriented as the rightmost two permutations shown in Figure \[fig-wedge-simples\]. This class was enumerated by Bóna [@bona:the-permutation:].

Grid Classes and Regular Languages {#sec-geom-grid-review}
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Given a permutation $\pi$ of length $n$ and sets $X,Y\subseteq[n]$, we write $\pi(X\times Y)$ for the permutation that is order isomorphic to the subsequence of $\pi$ with indices from $X$ and values in $Y$. For example, $286435179([4,9]\times[5,9])$ consists of the subsequence of entries in indices $4$ through $9$ which have values between $5$ and $9$; in this case the subsequence is $579$, so $286435179([4,9]\times[5,9])=123$.

Suppose that $M$ is a $t\times u$ matrix[^2]with entries from $\{0,\pm 1\}$. A *gridded permutation* is a permutation $\pi$ equipped with *row* and *column divisions* denoted respectively by $1=c_1\le\cdots\le c_{t+1}=n+1$ and $1=r_1\le\cdots\le r_{u+1}=n+1$ (where $n$ is the length of $\pi$). This gridded permutation (or simply, gridding of $\pi$) is *compatible* with the matrix $M$ (in which case we sometimes call it an $M$-gridding of $\pi$) if $\pi([c_k,c_{k+1})\times[r_\ell,r_{\ell+1}))$ is increasing whenever $M_{k,\ell}=1$, decreasing whenever $M_{k,\ell}=-1$, and empty whenever $M_{k,\ell}=0$. The [*(monotone) grid class of $M$*]{}, written ${\operatorname{Grid}}(M)$, consists of 