(0,0)(60,50) (0,10)(10,10)(10,20)(0,20) (0,20)(10,20)(10,30)(0,30) (0,30)(10,30)(10,40)(0,40) (0,40)(10,40)(10,50)(0,50) (10,0)(20,0)(20,10)(10,10) (10,40)(20,40)(20,50)(10,50) (10,40)(20,50) (20,0)(30,0)(30,10)(20,10) (20,30)(30,30)(30,40)(20,40) (20,40)(30,40)(30,50)(20,50) (30,0)(40,0)(40,10)(30,10) (30,40)(40,40)(40,50)(30,50) (40,0)(50,0)(50,10)(40,10) (40,10)(50,10)(50,20)(40,20) (40,20)(50,20)(50,30)(40,30) (40,30)(50,30)(50,40)(40,40) (50,10)(60,10)(60,20)(50,20) (50,20)(60,20)(60,30)(50,30) (50,30)(60,30)(60,40)(50,40) (10,20)[2.0]{} (10,20)[$d$]{} (20,40)[2.0]{} (20,40)[$b$]{} (30,10)[2.0]{} (30,10)[$a$]{} (40,30)[2.0]{} (40,30)[$c$]{} (50,50)[2.0]{} (0,0)(0,50) (0,0)(60,0) (10,0)(10,50) (0,10)(60,10) (20,0)(20,50) (0,20)(60,20) (30,0)(30,50) (0,30)(60,30) (40,0)(40,50) (0,40)(60,40) (50,0)(50,50) (0,50)(60,50) (60,0)(60,50)

(-3,0)(120,85) (0,0)(80,80) (80,40)(120,0) (80,40)(120,80) (0,0)(0,80) (40,0)(40,80) (80,0)(80,80) (120,0)(120,80) (0,0)(120,0) (0,40)(120,40) (0,80)(120,80) (-3,2)(-3,38) (-3,42)(-3,78) (2,83)(38,83) (42,83)(78,83) (82,83)(118,83) (35,35)[2.0]{} (90,30)[2.0]{} (25,25)[2.0]{} (100,60)[2.0]{} (105,15)[2.0]{} (110,70)[2.0]{} (75,75)[2.0]{} (40,0)[${\textsf{a}}$]{} (80,40)[${\textsf{b}}$]{} (80,0)[${\textsf{c}}$]{} (120,40)[${\textsf{d}}$]{}

We now consider the encoding ${\varphi}$ over the cell alphabet $\Sigma=\{{\textsf{a}},{\textsf{b}},{\textsf{c}},{\textsf{d}}\}$ as indicated in Figure \[fig-2473516\], which also shows an example of ${\varphi}$. This mapping is not injective on $\Sigma^\ast$ for the following two reasons.

-   The same gridded permutation may be the image of multiple words (in our example, this occurs because the pairs $\{{\textsf{a}},{\textsf{b}}\}$, $\{{\textsf{a}},{\textsf{d}}\}$, and $\{{\textsf{b}},{\textsf{c}}\}$ “commute”, i.e., they may be interchanged without affecting the gridded permutation obtained). A method to handle this issue in general (by appealing to the theory of “trace monoids”) is presented in [@albert:geometric-grid-: Section 7].

-   A given permutation may have several different $M$-griddings. A (nonconstructive) method to handle this issue in general is presented in [@albert:geometric-grid-: Section 8].

In the class we are considering, ${\operatorname{Av}}(4312,3142)$, it is possible to deal with the 