On Growth Rates of Closed Permutation Classes

A class of permutations $\Pi$ is called closed if $\pi\subset\sigma\in\Pi$ implies $\pi\in\Pi$, where the relation $\subset$ is the natural containment of permutations. Let $\Pi_n$ be the set of all permutations of $1,2,\dots,n$ belonging to $\Pi$. We investigate the counting functions $n\mapsto|\Pi_n|$ of closed classes. Our main result says that if $|\Pi_n| < 2^{n-1}$ for at least one $n\ge 1$, then there is a unique $k\ge 1$ such that $F_{n,k}\le |\Pi_n|\le F_{n,k}\cdot n^c$ holds for all $n\ge 1$ with a constant $c>0$. Here $F_{n,k}$ are the generalized Fibonacci numbers which grow like powers of the largest positive root of $x^k-x^{k-1}-\cdots-1$. We characterize also the constant and the polynomial growth of closed permutation classes and give two more results on these.