A survey of consecutive patterns in permutations

TL;DR

This survey reviews recent advances in the study of consecutive patterns in permutations, covering enumeration, classification, and applications to dynamical systems, highlighting the field's growth over the past 15 years.

Contribution

It provides a comprehensive overview of recent developments, techniques, and applications related to consecutive permutation patterns, emphasizing enumeration and classification.

Findings

Enumeration formulas for specific patterns

Classification into equivalence classes

Applications to dynamical systems

Abstract

A consecutive pattern in a permutation $\pi$ is another permutation $\sigma$ determined by the relative order of a subsequence of contiguous entries of $\pi$. Traditional notions such as descents, runs and peaks can be viewed as particular examples of consecutive patterns in permutations, but the systematic study of these patterns has flourished in the last 15 years, during which a variety of different techniques have been used. We survey some interesting developments in the subject, focusing on exact and asymptotic enumeration results, the classification of consecutive patterns into equivalence classes, and their applications to the study of one-dimensional dynamical systems.