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A permutation class $C$ is said to be splittable if there exist two propersubclasses $A, B \subsetneq C$ such that any $\sigma \in C$ can be red-bluecolored so that the red (respectively, blue) subsequence of $\sigma$ is orderisomorphic to an element of $A$ (respectively, $B$). The class $C$ is said tobe composable if there exists some number of proper subclasses $A_1, \dots, A_k\subsetneq C$ such that any $\sigma \in C$ can be written as $\alpha_1 \circ\dots \circ \alpha_k$ for some $\alpha_i \in A_i$. We answer a question ofKarpilovskij by showing that there exists a composable permutation class thatis not splittable. We also give a condition under which an infinite composableclass must be splittable.

The Relation between Composability and Splittability of Permutation Classes