Permutations avoiding sets of patterns with long monotone subsequences

Abstract

We enumerate permutations that avoid all but one of the k patterns of length k starting with a monotone increasing subsequence of length k−1. We compare the size of such permutation classes to the size of the class of permutations avoiding the monotone increasing subsequence of length k−1. In most cases, we determine the exponential growth rate of these permutation classes, while in the remanining cases, we present strong numerical evidence leading to a conjectured growth rate. We also present numerical evidence that suggests a conjecture for the growth rates of these permutation classes at subexponential precision. Some of these conjectures claim that the relevant permutation classes have non-algebraic, and in one case, even non-D-finite, generating functions.