Equivalence classes of mesh patterns with a dominating pattern

TL;DR

This paper studies the relationships between mesh patterns in permutations, providing conditions for when they are equivalent and classifying these equivalences within specific pattern families.

Contribution

It introduces sufficient conditions for mesh pattern coincidence and offers a complete classification of these patterns within certain permutation classes.

Findings

Complete classification of coincidences for length 2 mesh patterns with length 3 classical patterns.

Full Wilf-classification of length 2 mesh patterns in 231-avoiding permutations.

Conditions for pattern coincidence considering permutations avoiding longer classical patterns.

Abstract

Two mesh patterns are coincident if they are avoided by the same set of permutations, and are Wilf-equivalent if they have the same number of avoiders of each length. We provide sufficient conditions for coincidence of mesh patterns, when only permutations also avoiding a longer classical pattern are considered. Using these conditions we completely classify coincidences between families containing a mesh pattern of length 2 and a classical pattern of length 3. Furthermore, we completely Wilf-classify mesh patterns of length 2 inside the class of 231-avoiding permutations.