The poset of mesh patterns
Discrete Mathematics, Volume 343, Issue 6, June 2020
We introduce the poset of mesh patterns, which generalises the permutation
pattern poset. We fully classify the mesh patterns for which the interval [1,m]
is non-pure, where 1 is the unshaded singleton mesh pattern. We present some
results on the Möbius function of the poset, and show that its value on [1,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.