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 $\textbf{ 1423}$.
Categories: math.CO