The Poset of Mesh Patterns

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.