The Relation between Composability and Splittability of Permutation Classes

Abstract

A permutation class $C$ is said to be splittable if there exist two proper subclasses $A,B⊊C$ such that any $\sigma \in C$ can be red-blue colored so that the red (respectively, blue) subsequence of $\sigma$ is order isomorphic to an element of $A$ (respectively, $B$). The class $C$ is said to be composable if there exists some number of proper subclasses $A_1,\dots ,A_k⊊C$ such that any $\sigma \in C$ can be written as ${\alpha }_1\circ \cdots \circ {\alpha }_k$ for some ${\alpha }_i\in A_i$. We answer a question of Karpilovskij by showing that there exists a composable permutation class that is not splittable. We also give a condition under which an infinite composable class must be splittable.