Simple permutations: Decidability and unavoidable substructures

Simple permutations are the building blocks of permutation classes. As such, classes with only finitely many simple permutations, e.g., the class of 132-avoiding permutations, have nice properties. To name three: these classes have algebraic generating functions (as established by Albert and Atkinson [1]; see Brignall, Huczynska, and Vatter [9] for extensions), are partially well-ordered (see the conclusion), and are finitely based [1]. It is natural then to ask which finitely based classes contain only finitely many simple permutations. Our main result establishes that this can be done algorithmically: