# Automatic discovery of structural rules of permutation classes

## TL;DR

The paper presents an algorithm that automatically conjectures the structural rules of permutation classes, facilitating enumeration and random sampling, and extends to classes avoiding length four patterns.

## Contribution

It introduces a novel algorithm for discovering permutation class structures as disjoint rule covers, applicable to non-polynomial classes and aiding enumeration and sampling.

## Key findings

- Successfully applies to various permutation classes
- Enables enumeration based on discovered structures
- Facilitates uniform random sampling from classes

## Abstract

We introduce an algorithm that conjectures the structure of a permutation class in the form of a disjoint cover of "rules"; similar to generalized grid classes. The cover is usually easily verified by a human and translated into an enumeration. The algorithm is successful on different inputs than other algorithms and can succeed with any polynomial permutation class. We apply it to every non-polynomial permutation class avoiding a set of length four patterns. The structures found by the algorithm can sometimes allow an enumeration of the permutation class with respect to permutation statistics, as well as choosing a permutation uniformly at random from the permutation class. We sketch a new algorithm formalizing the human verification of the conjectured covers.