On the effective and automatic enumeration of polynomial permutation classes

The “Fibonacci Dichotomy” of Kaiser and Klazar [15] was one of the first general results on the enumeration of permutation classes. It states that if there are fewer permutations of length n in a permutation class than the nth Fibonacci number, for any n, then the enumeration of the class is given by a polynomial for sufficiently large n. Since the Fibonacci Dichotomy was established for permutation classes, Balogh, Bollobás, and Morris [6] showed that it extends to the (more general) context of ordered graphs, while other proofs of the Fibonacci Dichotomy for permutations have been given by Huczynska and Vatter [14] and Albert, Atkinson, and Brignall [3].