Mesh patterns with superfluous mesh

TL;DR

This paper characterizes mesh patterns that do not add complexity beyond classical permutation patterns, identifying those with minimal and maximal superfluous meshes and their enumeration.

Contribution

It provides a complete description of mesh patterns with avoidance coincident with classical patterns and analyzes permutations with extreme numbers of superfluous meshes.

Findings

Identifies all mesh patterns with avoidance coincident with classical patterns.

Characterizes permutations with the fewest and most superfluous meshes.

Enumerates permutations based on their superfluous mesh counts.

Abstract

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.