Algorithmic coincidence classification of mesh patterns

Permutations have connections to other mathematical objects such as Schubert varieties, sorting networks, and genome rearrangements. Often the connection is described in terms of patterns that are absent from the permutations. There can be ambiguity in this description, in the sense that the same subset of permutations can be defined with two different patterns. This is called coincidence and we focus on the coincidence of mesh patterns, one of the most descriptive version of patterns in permutations. We review and extend previous results on coincidence of mesh patterns. We introduce the notion of a force on a permutation pattern and apply it to the coincidence classification of mesh patterns, completing the classification up to size three. We also show that this concept can be used to enumerate classical permutation classes. •We present a new algorithm for the coincidence classification of mesh patterns.•We complete the coincidence classification of mesh patterns of length 3.•We use our results to count some permutation classes.