Enumeration of several two-by-four classes

We use catalytic variables to derive generating functions for the permutation classes $Av(\textbf{4123},\textbf{1324})$, $Av(\textbf{4123},\textbf{1243})$, and $Av(\textbf{4123},\textbf{1342})$. Each generating function is algebraic of degree two, and the growth rates of the classes are 4, 5, and $\alpha$, respectively, where $\alpha \approx 4.17035$. As a consequence of our analysis, we see that a typical large permutation in $Av(\textbf{4123},\textbf{1324})$ is likely to also avoid $\textbf{3124}$, and is even more likely to avoid $\textbf{31524}$. Large permutations which avoid $\textbf{4123}$ and $\textbf{1243}$ are likely to also avoid $\textbf{1423}$.