Unsplittable classes of separable permutations

TL;DR

This paper characterizes the structurally and basally defined unsplittable subclasses within the class of separable permutations, enhancing understanding of permutation class decompositions.

Contribution

It provides a complete structural and basis-based characterization of unsplittable subclasses of separable permutations, a previously unresolved problem.

Findings

Identifies the structural properties of unsplittable subclasses.

Provides basis characterizations for these subclasses.

Advances the theory of permutation class decompositions.

Abstract

A permutation class is splittable if it is contained in the merge of two of its proper subclasses. We characterise the unsplittable subclasses of the class of separable permutations both structurally and in terms of their bases.