# Enumeration of small Wilf classes avoiding 1324 and two other 4-letter patterns

## TL;DR

This paper advances the enumeration of permutation classes avoiding certain patterns by deriving generating functions for most triples containing 1324, using combinatorial and analytic methods like generating trees and kernel method.

## Contribution

It provides explicit generating functions for nearly all triples containing 1324, except one conjectured intractable case, enhancing understanding of Wilf classes.

## Key findings

- Generated functions for most triples containing 1324
- Used combinatorial and analytic methods such as generating trees and kernel method
- Identified an intractable triple conjecture

## Abstract

Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutation patterns by showing that there are 32 non-singleton Wilf classes. Moreover, the generating function for each triple lying in a non-singleton Wilf class has been explicitly determined. In this paper, toward the goal of enumerating avoiders for the singleton Wilf classes, we obtain the generating function for all but one of the triples containing 1324. (The exceptional triple is conjectured to be intractable.) Our methods are both combinatorial and analytic, including generating trees, recurrence relations, and decompositions by left-right maxima. Sometimes this leads to an algebraic equation for the generating function, sometimes to a functional equation or a multi-index recurrence amenable to the kernel method.