The most and the least avoided consecutive patterns

Abstract

We prove that the number of permutations avoiding the consecutive pattern 12… m, that is, containing no m adjacent entries in increasing order, is asymptotically larger than the number of permutations avoiding any other consecutive pattern of length m. This settles a conjecture of Elizalde and Noy from 2001. We also prove a recent conjecture of Nakamura stating that, at the other end of the spectrum, the number of permutations avoiding 12… (m−2)m(m−1) is asymptotically smaller than for any other pattern. Finally, we consider non-overlapping patterns and obtain analogous results describing the most and least avoided ones. The techniques used include the cluster method of Goulden and Jackson, an interpretation of clusters as linear extensions of posets, and singularity analysis of generating functions.