the class is [*sum closed*]{}. Thus, letting $f$ denote the generating function for nonempty permutations in ${\operatorname{Av}}(4213,3142)$ and $f_\oplus$ denote the generating function for sum decomposable permutations, we see that $f_\oplus=\left(f-f_\oplus\right)f$, from which it follows that $$f_\oplus=\frac{f^2}{1+f}.$$ For skew sums, we have that $\pi\ominus\sigma\in{\operatorname{Av}}(4213,3142)$ if and only if $\pi\in{\operatorname{Av}}(4213,3142)$ and $\sigma\in{\operatorname{Av}}(213)$. Letting $$c=\frac{1-2x-\sqrt{1-4x}}{2x}$$ denote the generating function for the Catalan numbers (with constant term zero), which is well-known as the generating function of nonempty permutations in ${\operatorname{Av}}(213)$, we have $f_\ominus=\left(f-f_\ominus\right)c$, so $$f_\ominus=\frac{cf}{1+c}.$$

Now we must count the inflations of the parallel alternations $246\cdots (2m)135\cdots (2m-1)$ for $m\ge 2$. This is relatively straightforward:

-   the interval inflating $2m-1$ must avoid $213$,

-   all other intervals inflating odd entries must be increasing, and

-   even entries may be inflated by any element of ${\operatorname{Av}}(4213,3142)$.

This shows that the contribution of inflations of $246\cdots (2m)135\cdots (2m-1)$, for each $m\ge 2$, is $$f^m\left(\frac{x}{1-x}\right)^{m-1}c,$$ showing that $$f
=
x+f_\oplus+f_\ominus+\sum_{m=2}^\infty f^m\left(\frac{x}{1-x}\right)^{m-1}c
=
x+\frac{f^2}{1+f}+\frac{cf}{1+c}+\frac{xcf^2}{1-x-xf}.$$ From this we obtain:

\[thm-3142-4213\] The generating function $f$ for ${\operatorname{Av}}(4213,3142)$ satisfies $$\begin{array}{rclcc}
x^3f^6
&+&
(7x^3-7x^2+2x)f^5&&\\
&+&
(x^4+14x^3-21x^2+10x-1)f^4&&\\
&+&
(4x^4+8x^3-19x^2+11x-2)f^3&&\\
&+&
(6x^4-5x^3-2x^2+2x)f^2&&\\
&+&
(4x^4-7x^3+4x^2-x)f&&\\
&+&
x^4-2x^3+x^2
&=&0.
\end{array}$$

The first several terms of this sequence are $$1,2,6,22,89,379,1664,7460,33977,156727,730619,3436710,16291842,77758962,$$ sequence [A165541]{} in the [OEIS [@sloane:the-on-line-enc:]]{}. 