Rationality for subclasses of 321-avoiding permutations

TL;DR

This paper proves that certain subclasses of 321-avoiding permutations have rational generating functions by establishing a bijection with regular languages, using novel encodings like the panel encoding.

Contribution

It introduces a new panel encoding and demonstrates that subclasses defined by finite restrictions or well quasi-ordering correspond to regular languages.

Findings

Proper subclasses have rational generating functions.

Subclasses are bijective with regular languages.

Introduces the novel panel encoding method.

Abstract

We prove that every proper subclass of the 321-avoiding permutations that is defined either by only finitely many additional restrictions or is well quasi-ordered has a rational generating function. To do so we show that any such class is in bijective correspondence with a regular language. The proof makes significant use of formal languages and of a host of encodings, including a new mapping called the panel encoding that maps languages over the infinite alphabet of positive integers avoiding certain subwords to languages over finite alphabets.