  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.
