The limit of a Stanley–Wilf sequence is not always rational, and layered patterns beat monotone patterns

We show the first known example for a pattern q for which is not an integer, where S_n(q) denotes the number of permutations of length n avoiding the pattern q. We find the exact value of the limit and show that it is irrational, but algebraic. Then we generalize our results to an infinite sequence of patterns. We provide further generalizations that start explaining why certain patterns are easier to avoid than others. Finally, we show that if q is a layered pattern of length k, then L(q)(k-1)² holds.