Enumeration of several two-by-four classes

TL;DR

This paper derives algebraic generating functions for specific permutation classes avoiding certain patterns, revealing their growth rates and typical structure of large permutations within these classes.

Contribution

It introduces a method using catalytic variables to explicitly compute algebraic generating functions for several permutation classes.

Findings

Generating functions are algebraic of degree two.

Growth rates are 4, 5, and approximately 4.17035.

Large permutations tend to also avoid certain additional patterns.

Abstract

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 $\textbf{1423}$.