Coincidence among families of mesh patterns

The Australasian Journal of Combinatorics 2015, Volume 63 Part 1 (2015)

Anders, Henning and Bridget

Two mesh patterns are coincident if they are avoided by the same set of permutations. In this paper, we provide necessary conditions for this coincidence, which include having the same set of enclosed diagonals. This condition is sufficient to prove coincidence of vincular patterns, although it is not enough to guarantee coincidence of bivincular patterns. In addition, we provide a generalization of the Shading Lemma (Hilmarsson et al.), a result that examined when a square could be added to the mesh of a pattern.

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• The Australasian Journal of Combinatorics 2015, Volume 63 Part 1 (2015)
• arXiv

Presentations

Later work (last updated 3 October 2026)

• M. Tannock and H. Úlfarsson, Equivalence classes of mesh patterns with a dominating pattern, Discrete Math. Theor. Comput. Sci. 19 (2018). Extends the classification of coincidences among mesh patterns of length 2 given here to permutations that also avoid a classical pattern of length 3.
• D. Govc and J. P. Smith, Asymptotic behaviour of the containment of certain mesh patterns, Discrete Math. 345 (2022). Results on the proportion of permutations that contain certain mesh patterns, using the Simultaneous Shading Lemma of this paper.
• C. Bean, B. Gudmundsson, T. K. Magnússon and H. Úlfarsson, Algorithmic coincidence classification of mesh patterns, Inform. and Comput. 292 (2023). Generalizes the Simultaneous Shading Lemma with forces, and turns the classification of coincidences into an algorithm.
• S. Kitaev, D. Qiu and C. Xu, Coincidences and growth of boxed mesh patterns, preprint (2026). Classifies the coincidences of boxed mesh patterns with classical and vincular patterns, using the enclosed-diagonal condition of this paper.
• All citing papers on Google Scholar (11)