On Growth Rates of Closed Permutation Classes

A class of permutations Π is called closed if π ⊂ σ ∈ Π implies π ∈ Π, where the relation ⊂ is the natural containment of permutations. Let Πn be the set of all permutations of 1, 2, . . . , n belonging to Π. We investigate the counting functions n 7→ |Πn| of closed classes. Our main result says that if |Πn| < 2n−1 for at least one n ≥ 1, then there is a unique k ≥ 1 such that Fn,k ≤ |Πn| ≤ Fn,k · nc holds for all n ≥ 1 with a constant c > 0. Here Fn,k are the generalized Fibonacci numbers which grow like powers of the largest positive root of xk − xk−1 − · · · − 1. We characterize also the constant and the polynomial growth of closed permutation classes and give two more results on these.