the words of length at least four which begin with ${\textsf{c}}$ have multivariate generating function ${x_{\textsf{c}}}s({x_{\textsf{a}}},{x_{\textsf{b}}},{x_{\textsf{c}}},{x_{\textsf{d}}})$. Note that our rules preclude words encoding simple permutations from beginning with ${\textsf{b}}$ or ${\textsf{d}}$.

Now we characterise the inflations. Because $3142$ is simple, it will not occur when inflating a $3142$-avoiding permutation by $3142$-avoiding intervals, so we need only avoid $4312$. Since the class is sum closed, we have that $f_\oplus=f^2/(1+f)$, as in Section \[sec-4213-3142\]. The skew decomposable permutations are a bit more complicated, but divide into a union: $$\left({\operatorname{Av}}(21)\ominus{\operatorname{Av}}(312)\right)\cup\left({\operatorname{Av}}_{\not\ominus}(4312,3142)\ominus{\operatorname{Av}}(12)\right),$$ where ${\operatorname{Av}}_{\not\ominus}(4312,3142)$ denotes the set of skew indecomposable permutations in this class. As the intersection of these two is simply ${\operatorname{Av}}(21)\ominus{\operatorname{Av}}(12)$, $f_\ominus=mc+(f-f_\ominus)m-m^2$, where $$m=\frac{x}{1-x}$$ denotes the generating function for the nonempty decreasing (or, increasing) permutations. Solving this shows $$f_\ominus
=
\frac{m(f+c-m)}{1+m}.$$ Inflations of simple permutation of length at least four are a bit more complicated, as there are several cases. In all such inflations, each entry which corresponds to a ${\textsf{b}}$ may only be inflated by an increasing permutation (but may be inflated by any such permutation), while each entry which corresponds to a ${\textsf{d}}$ may only be inflated by a $312$-avoiding permutation (but may be inflated by any such permutation). If the 