Subclasses of the separable permutations

TL;DR

This paper proves that certain subclasses of separable permutations have rational generating functions, using new tools like strongly rational classes, partial well-order, and atomicity.

Contribution

Introduces the concept of strongly rational permutation classes and applies it to prove rational generating functions for subclasses of separable permutations.

Findings

All subclasses of separable permutations excluding Av(231) or its symmetries have rational generating functions.

Develops the theory of strongly rational permutation classes for the first time.

Utilizes partial well-order and atomicity in permutation class analysis.

Abstract

We prove that all subclasses of the separable permutations not containing Av(231) or a symmetry of this class have rational generating functions. Our principal tools are partial well-order, atomicity, and the theory of strongly rational permutation classes introduced here for the first time.