(20,10)[0.04in]{} (30,60)[0.04in]{} (40,20)[0.04in]{} (50,70)[0.04in]{} (60,30)[0.04in]{} (70,80)[0.04in]{} (80,40)[0.04in]{}

&

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&

(0,0)(80,80) (0,0)(80,80) (10,70)[0.04in]{} (20,50)[0.04in]{} (30,30)[0.04in]{} (40,10)[0.04in]{} (50,80)[0.04in]{} (60,60)[0.04in]{} (70,40)[0.04in]{} (80,20)[0.04in]{}

&

------------------------------------------------------------------------

&

(0,0)(80,80) (0,0)(80,80) (10,40)[0.04in]{} (20,80)[0.04in]{} (30,30)[0.04in]{} (40,70)[0.04in]{} (50,20)[0.04in]{} (60,60)[0.04in]{} (70,10)[0.04in]{} (80,50)[0.04in]{}

Schmerl and Trotter [@schmerl:critically-inde: Corollary 5.10] proved a result (in the more general context of irreflexive binary relational structures) which in our context states that every simple permutation of length $n$ which is not a parallel alternation contains simple subpermutations of every length $5\le m\le n$. Therefore, in order to establish that the permutation class ${\mathcal{C}}$ contains *only* parallel alternations, we just need to check that it does not contain any simple permutation of length $5$, i.e., that $${\mathcal{C}}\subseteq{\operatorname{Av}}(24153, 25314, 31524, 35142, 41352, 42513).$$ Clearly this holds for the class ${\operatorname{Av}}(4213,3142)$, because $4213$ is contained in $25314$ and $42513$ while $3142$ is contained in $24153$, $31524$, $35142$, and $41352$. Moreover, it is easily seen (because of the basis element $3142$) that ${\operatorname{Av}}(4213,3142)$ can contain only parallel alternations oriented as on the left of Figure \[fig-par-alts\], i.e., those of the form $$246\cdots(2m)135\cdots(2m-1).$$

With the simple permutations characterised, we now describe the allowed inflations. It is easy to see that $\pi\oplus\sigma\in{\operatorname{Av}}(4213,3142)$ for all $\pi,\sigma\in{\operatorname{Av}}(4213,3142)$, or in other words, that 