A probabilistic approach to consecutive pattern avoiding in permutations

TL;DR

This paper introduces a probabilistic method for counting permutations avoiding specific consecutive patterns, providing bounds, proving the CMP conjecture, and analyzing pattern avoidance behavior.

Contribution

It offers a new probabilistic framework for pattern avoidance enumeration and proves the CMP conjecture with insights into pattern avoidance tendencies.

Findings

Established explicit bounds on pattern-avoiding permutations

Provided a simple proof of the CMP conjecture

Showed most patterns behave like the least avoided pattern

Abstract

We present a new approach to the problem of enumerating permutations of length n that avoid a fixed consecutive pattern of length m. We use this idea to give explicit upper and lower bounds on the number of permutations avoiding a pattern of length m. As a corollary, we obtain a simple proof of the CMP conjecture, regarding the most avoided pattern, recently shown by Elizalde. Finally, we also show that most of the patterns behave similar to the least avoided one.