# Uniquely-Wilf classes

## TL;DR

This paper characterizes infinite permutation classes where each class contains only one Wilf-equivalence class per size, under certain avoidance conditions, revealing structural properties of these uniquely-Wilf classes.

## Contribution

It provides a characterization of infinite classes with unique Wilf-equivalence classes per size, given avoidance of specific small permutations.

## Key findings

- Characterization of infinite uniquely-Wilf classes with avoidance conditions
- Identification of structural properties of these classes
- Extension of Wilf-equivalence concepts to broader classes

## Abstract

Two permutations in a class are Wilf-equivalent if, for every size, $n$, the number of permutations in the class of size $n$ containing each of them is the same. Those infinite classes that have only one equivalence class in each size for this relation are characterised provided either that they avoid at least one permutation of size 3, or at least three permutations of size 4.