Equidistribution of mesh patterns of short length

TL;DR

This paper classifies mesh patterns of length 2 based on their equidistribution properties, reducing the number of Wilf-classes and solving open enumeration problems using combinatorial and algebraic techniques.

Contribution

It provides a near-complete classification of length-2 mesh patterns, introduces new equidistribution results, and improves bounds on Wilf-classes.

Findings

Number of equidistribution classes between 105 and 108, conjectured to be 105.

Upper bound of 49 Wilf-classes, improved from 56.

Resolved seven open pattern-avoidance enumeration problems.

Abstract

We study the equidistribution of mesh patterns of length 2. We show that the number of equidistribution equivalence classes lies between 105 and 108, and conjecture that it is exactly 105. As a consequence, we obtain an upper bound of 49 Wilf-classes, improving the previously known bound of 56 due to Hilmarsson et al., and reducing the problem to three remaining conjectural equivalences (with the actual number conjectured to be 46). Our approach combines bijective constructions, generating functions, recurrence relations, and structural symmetries. We establish several new equidistribution results, including four previously unknown distribution classes, connect numerous patterns to known distributions in the literature, and resolve seven open pattern-avoidance enumeration problems posed by Hilmarsson et al. This work provides a near-complete classification of mesh patterns of length 2 and unifies several previously isolated results within a coherent framework.