issues concretely.

First we address (G1). For any particular gridded permutation, we prefer the lexicographically minimal word encoding it. For example, suppose that a word contained a factor of the form $\{{\textsf{b}},{\textsf{d}}\}^+{\textsf{a}}$ (here the $+$ superscript signifies that this portion of the word contains at least one letter). We could then replace this factor by a factor of the form ${\textsf{a}}\{{\textsf{b}},{\textsf{d}}\}^+$ and obtain a lexicographically lesser word which is mapped to the same permutation. Therefore we forbid factors of the form $\{{\textsf{b}},{\textsf{d}}\}^+{\textsf{a}}$. The other factor we need to forbid is ${\textsf{c}}{\textsf{a}}^\ast{\textsf{b}}$ (which could be replaced by a factor of the form ${\textsf{b}}{\textsf{c}}{\textsf{a}}^\ast$).

Now we address (G2), which requires us to choose a preferred (geometric) $M$-gridding for every permutation in ${\operatorname{Geom}}(M)$. Among all $M$-griddings of a permutation, we prefer the one that has the most entries in the first column, then the most entries in the second column, and then the most entries in the first row. Thus in terms of column divisions $1=c_1\le c_2\le c_3\le c_4=n+1$ and row divisions $1=r_1\le r_2\le r_3=n+1$, we seek to maximise $c_2$, then $c_3$, and then $r_2$. The words which correspond to such griddings can now be characterised as those which do not begin with 