The Relation between Composability and Splittability of Permutation Classes
英文摘要：
A permutation class $C$ is said to be splittable if there exist two proper subclasses $A, B \subsetneq C$ such that any $σ\in C$ can be red-blue colored so that the red (respectively, blue) subsequence of $σ$ 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 $A_1, \dots, A_k \subsetneq C$ such that any $σ\in C$ can be written as $α_1 \circ \dots \circ α_k$ for some $α_i \in A_i$. 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.
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