all permutations which possess a gridding compatible with $M$. Figure \[fig-grid-geom-three-pane\] shows an example.

As illustrated by Murphy and Vatter [@murphy:profile-classes:], monotone grid classes can display chaotic and unstructured behaviour. However, it has recently been shown that these classes contain subclasses with especially amenable structure. To define these subclasses, consider the point set in $\mathbb{R}^2$ (called the *standard figure* of the ${0/\mathord{\pm} 1}$ matrix $M$) consisting of cells $C_{kl}$ whose contents are:

-   the line segment from $(k-1,\ell-1)$ to $(k,\ell)$ if $M_{k,\ell}=1$ or

-   the line segment from $(k-1,\ell)$ to $(k,\ell-1)$ if $M_{k,\ell}=-1$ or

-   empty if $M_{kl}=0$.

The *geometric grid class* of $M$, denoted by ${\operatorname{Geom}}(M)$, is then the set of all permutations that can be drawn on this figure in the following manner. Choose $n$ points in the figure, no two on a common horizontal or vertical line. Then label the points from $1$ to $n$ from bottom to top and record these labels reading left to right. The centre pane of Figure \[fig-grid-geom-three-pane\] shows a permutation from a geometric grid class, while the right pane demonstrates that $2413$ is not in this geometric grid class.

It sometimes happens that ${\operatorname{Grid}}(M)={\operatorname{Geom}}(M)$; to characterise this phenomenon, we need to 