The Poset of Mesh Patterns

TL;DR

This paper introduces the poset of mesh patterns, classifies certain non-pure intervals, analyzes the M"obius function, and explores disconnected and non-shellable intervals through a generalization of permutation operations.

Contribution

It generalizes the permutation pattern poset to mesh patterns and provides a classification of non-pure intervals and properties of the M"obius function.

Findings

Classification of non-pure intervals in the mesh pattern poset

Most M"obius function values are zero for the unshaded singleton mesh pattern

Introduction of disconnected and non-shellable intervals via a generalized product operation

Abstract

We introduce the poset of mesh patterns, which generalises the permutation pattern poset. We fully classify the mesh patterns for which the interval [1^\emptyset,m] is non-pure, where 1^\emptyset is the unshaded singleton mesh pattern. We present some results on the M\"obius function of the poset, and show that {\mu}(1^\emptyset,m) is almost always zero. Finally, we introduce a class of disconnected and non-shellable intervals by generalising the direct product operation from permutations to mesh patterns.