Counting occurrences of patterns in permutations

TL;DR

This paper introduces a new method for counting pattern occurrences in permutations, extending known results for length-3 patterns and exhaustively analyzing length-4 patterns, leading to conjectures on their asymptotic behaviors.

Contribution

A novel counting algorithm for permutation patterns, extended analysis of length-3 patterns, and comprehensive enumeration and conjectures for length-4 patterns.

Findings

Confirmed two Wilf classes for length-3 patterns

Identified seven Wilf classes for length-4 patterns

Conjectured asymptotic behaviors for all length-4 classes

Abstract

We develop a new, powerful method for counting elements in a multiset. As a first application, we use this algorithm to study the number of occurrences of patterns in a permutation. For patterns of length 3 there are two Wilf classes, and the general behaviour of these is reasonably well-known. We slightly extend some of the known results in that case, and exhaustively study the case of patterns of length 4, about which there is little previous knowledge. For such patterns, there are seven Wilf classes, and based on extensive enumerations and careful series analysis, we have conjectured the asymptotic behaviour for all classes.