Juxtaposing Catalan permutation classes with monotone ones

TL;DR

This paper systematically enumerates specific permutation classes formed by juxtaposing Catalan classes with monotone classes, using Dyck paths and context-free grammars for enumeration.

Contribution

It introduces a method to enumerate all such juxtaposition classes of permutations using Dyck paths decorated with sequences and context-free grammars.

Findings

Complete enumeration of all juxtaposition classes of the specified form

Development of a Dyck path-based representation for these classes

Application of context-free grammars for enumeration

Abstract

This paper enumerates all juxtaposition classes of the form "Av($abc$) next to Av($xy$)", where $abc$ is a permutation of length three and $xy$ is a permutation of length two. We use Dyck paths decorated by sequences of points to represent elements from such a juxtaposition class. Context-free grammars are then used to enumerate these decorated Dyck paths.