The most and the least avoided consecutive patterns

TL;DR

This paper determines which consecutive patterns are most and least avoided in permutations, settling longstanding conjectures by analyzing asymptotic counts for pattern avoidance.

Contribution

It proves the extremal avoidance counts for specific patterns, confirming conjectures and advancing understanding of permutation pattern avoidance asymptotics.

Findings

Maximal avoidance for pattern 12...m

Minimal avoidance for pattern 12...(m-2)m(m-1)

Largest avoidance among non-overlapping patterns is 134...m2

Abstract

We prove that the number of permutations avoiding an arbitrary consecutive pattern of length m is asymptotically largest when the avoided pattern is 12...m, and smallest when the avoided pattern is 12...(m-2)m(m-1). This settles a conjecture of the author and Noy from 2001, as well as another recent conjecture of Nakamura. We also show that among non-overlapping patterns of length m, the pattern 134...m2 is the one for which the number of permutations avoiding it is asymptotically largest.