On growth rates of closed permutation classes

Abstract

A class of permutations ∏ is called closed if π ⊂ σ ∈ ∏ implies π ∈ ∏, where the relation ⊂ is the natural containment of permutations. Let ∏n be the set of all permutations of 1, 2, . . ., n belonging to ∏. We investigate the counting functions n → |∏n| of closed classes. Our main result says that if |∏n| < 2n-1 for at least one n ≥ 1, then there is a unique k ≥ 1 such that Fn,k ≤ |∏n | ≤ 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 x k - xk -1- ⋯ -1. We characterize also the constant and the polynomial growth of closed permutation classes and give two more results on these.