Finding regular insertion encodings for permutation classes

Permutation classes, or restricted permutations, have received considerable attention over the past two decades, and during this time a great variety of techniques have been used to enumerate them. One of the most popular approaches, pioneered by Chung, Graham, Hoggatt, and Kleiman [4], employs generating trees. The permutation classes with finitely labeled generating trees were characterized in Vatter [15]. A more powerful technique based on formal languages and called the insertion encoding was later introduced by Albert, Linton, and Ruškuc [2]. While they characterized the classes that possess regular insertion encodings, naively employing their techniques requires the determinization of non-deterministic automata several times, and no implementation has been available. We study regular insertion encodings from a new point of view, essentially focusing on accepting automata instead of languages. This leads both to an implementation (the Maple package INSENC, available for download from the author’s homepage) and to a new proof of the characterization of permutation classes with regular insertion encodings. We begin with definitions. Two sequences of natural numbers are said to be order isomorphic if they have the same pairwise comparisons, so 9, 1, 6, 7, 2 is order isomorphic to 5, 1, 3, 4, 2. Every sequence w of natural numbers without repetition is order isomorphic