The discriminant of the polynomial defining the generating function has a smallest positive root $\rho \approx 0.1895$, which is therefore the radius of convergence of the generating function and as ${\operatorname{Av}}(4213,3142)$ is sum closed (and hence the sequence $f_n$ is supermultiplicative) we can conclude that $f_n^{1/n} \to 1/\rho \approx 5.2778$. More detailed on the asymptotic behaviour of $f_n$ could be determined by standard methods as found for instance in Flajolet and Sedgewick [@flajolet:analytic-combin: Section VII.7].

[ccccccc]{}

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

&

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&

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

&

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

&

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

This is not the only 2 $\times$ 4 class to which such elementary techniques apply. For example:

-   ${\operatorname{Av}}(4213,1342)$ contains precisely two simple permutations of each length $n\ge 4$, both of which are wedge simple permutations oriented as the first two permutations shown in Figure \[fig-wedge-simples\]. This family of simple permutations is well enough behaved that we could enumerate the class, but this has already been done by Kremer and Shiu [@kremer:finite-transiti:] and can now 