The poset of mesh patterns

Abstract

We introduce the poset of mesh patterns, which generalizes the permutation pattern poset. We fully classify the mesh patterns for which the interval [10̸,m] is non-pure, where 10̸ is the unshaded singleton mesh pattern. We present some results on the Möbius function of the poset, and show that μ(10̸,m) is almost always zero. Finally, we introduce a class of disconnected and non-shellable intervals by generalizing the direct product operation from permutations to mesh patterns.