Wilf-classification of mesh patterns of short length

TL;DR

This paper initiates the Wilf-classification of length-2 mesh patterns, reducing the initial 1024 patterns to an estimated 46 classes using automated methods and enumeration.

Contribution

It introduces automated techniques to classify mesh patterns of length 2 and narrows down the number of Wilf-classes significantly.

Findings

Reduced potential Wilf-classes from 1024 to 56

Enumerated several classes to refine the count

Conjecture that the total number of classes is 46

Abstract

This paper starts the Wilf-classification of mesh patterns of length 2. Although there are initially 1024 patterns to consider we introduce automatic methods to reduce the number of potentially different Wilf-classes to at most 65. By enumerating some of the remaining classes we bring that upper-bound further down to 56. Finally, we conjecture that the actual number of Wilf-classes of mesh patterns of length 2 is 46.