Splittability and 1-amalgamability of permutation classes

TL;DR

This paper investigates the properties of permutation classes related to splittability and 1-amalgamability, demonstrating that these properties are not equivalent and introducing LR-inflations as a new concept.

Contribution

It proves that splittability and 1-amalgamability are not equivalent by analyzing the class Av(1423, 1342) and introduces LR-inflations, a novel concept of independent interest.

Findings

Unsplittability does not imply 1-amalgamability.

The class Av(1423, 1342) is both splittable and 1-amalgamable.

LR-inflations are introduced as a new tool for permutation class analysis.

Abstract

A permutation class $C$ is splittable if it is contained in a merge of two of its proper subclasses, and it is 1-amalgamable if given two permutations $\sigma$ and $\tau$ in $C$, each with a marked element, we can find a permutation $\pi$ in $C$ containing both $\sigma$ and $\tau$ such that the two marked elements coincide. It was previously shown that unsplittability implies 1-amalgamability. We prove that unsplittability and 1-amalgamability are not equivalent properties of permutation classes by showing that the class $Av(1423, 1342)$ is both splittable and 1-amalgamable. Our construction is based on the concept of LR-inflations, which we introduce here and which may be of independent interest.