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Mesh patterns and the expansion of permutation statistics as sums of permutation patterns

<12>The significance of this paper was the introduction of a new type of permutation pattern, mesh patterns, that unifies most others in the literature. Further significance was the strikingly novel result, called the ``Reciprocity Theorem'', relating a mesh pattern algebraically to its dual. Mesh patterns are having a major impact, e.g. Ulfarsson (2012) characterised 3-stack sortable permutations via mesh patterns (open since 1990). They have also helped characterise certain Schubert varieties (arXiv:1111.6146). Five of 37 talks at the 2012 permutation patterns conference included mesh patterns in an essential way, making this work agenda-setting.