Emerging consecutive pattern avoidance

TL;DR

This paper investigates the asymptotic likelihood of finding specific consecutive patterns in permutations avoiding certain length-3 patterns, revealing structural insights and open problems in permutation pattern avoidance.

Contribution

It provides a detailed analysis of pattern popularity in multiple classes of pattern-avoiding permutations, combining analytical and bijective methods, and identifies unresolved cases.

Findings

Popularity determined for ten classes based on permutation structure

Detailed analysis of two complex cases using combined methods

Open problem remains for five classes

Abstract

In this note we study the {\em asymptotic popularity}, that is, the limit probability to find a given consecutive pattern at a random position in a random permutation in the eighteen classes of permutations avoiding at least two length 3 consecutive patterns. We show that for ten classes, this popularity can be readily deduced from the structure of permutations. By combining analytical and bijective approaches, we study in details two more involved cases. The problem remains open for five classes.