The enumeration of three pattern classes

TL;DR

This paper determines the structure and enumerates three specific permutation classes using monotone grid classes, diagrams, and regular languages, resulting in rational generating functions.

Contribution

It introduces a method to analyze complex permutation classes via monotone grid classes, providing explicit structural descriptions and enumeration formulas.

Findings

Structures of three permutation classes are characterized.

Rational generating functions for these classes are derived.

Permutation classes are described using simple diagrams and regular languages.

Abstract

The structure of three pattern classes Av(2143, 4321), Av(2143, 4312) and Av(1324, 4312) is determined using the machinery of monotone grid classes. This allows the permutations in these classes to be described in terms of simple diagrams and regular languages and, using this, the rational generating functions which enumerate these classes are determined.