# Composability of Permutation Classes

## TL;DR

This paper investigates the composability of permutation classes through the operation of composition, analyzing their structure and providing examples of classes with varying composability properties.

## Contribution

It introduces the concept of composing permutation classes and characterizes classes based on their ability to be formed from proper subclasses.

## Key findings

- Some classes can be composed from two proper subclasses.
- Other classes require three subclasses for composition.
- Certain classes cannot be composed from any finite number of proper subclasses.

## Abstract

We define the operation of composing two hereditary classes of permutations using the standard composition of permutations as functions and we explore properties and structure of permutation classes considering this operation. We mostly concern ourselves with the problem of whether permutation classes can be composed from their proper subclasses. We provide examples of classes which can be composed from two proper subclasses, classes which can be composed from three but not from two proper subclasses and classes which cannot be composed from any finite number of proper subclasses.