Mesh patterns with superfluous mesh

Mesh patterns are a generalization of classical permutation patterns that encompass classical, bivincular, Bruhat-restricted patterns, and some barred patterns. In this paper, we describe all mesh patterns whose avoidance is coincident with classical avoidance, in a sense declaring that the additional data of a mesh was unnecessary for these patterns. We also describe the permutations having the fewest superfluous meshes, and the permutations having the most, enumerating the superfluous meshes in each case.