Asymptotic enumeration of permutations avoiding generalized patterns

Abstract

Motivated by the recent proof of the Stanley–Wilf conjecture, we study the asymptotic behavior of the number of permutations avoiding a generalized pattern. Generalized patterns allow the requirement that some pairs of letters must be adjacent in an occurrence of the pattern in the permutation, and consecutive patterns are a particular case of them. We determine the asymptotic behavior of the number of permutations avoiding a consecutive pattern, showing that they are an exponentially small proportion of the total number of permutations. For some other generalized patterns we give partial results, showing that the number of permutations avoiding them grows faster than for classical patterns but more slowly than for consecutive patterns.