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Quadrant marked mesh patterns in 132-avoiding permutations

Kitaev, Sergey and Remmel, Jeffrey and Tiefenbruck, Mark (2012) Quadrant marked mesh patterns in 132-avoiding permutations. Pure Mathematics and Applications, 23 (3). pp. 219-256. ISSN 1218-4586

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This paper is a continuation of the systematic study of the distributions of simple marked mesh patterns initiated in [6]. We study simple marked mesh patterns on 132-avoiding permutations. We derive generating functions for the number of occurrences of 4-parameter simple marked mesh patterns where only one parameter is allowed to be non-zero or a non-empty set. We show that specializations of such generating functions count a number of classical combinatorial sequences. Generating functions for the number of occurrences of 4-parameter simple marked mesh patterns where two or more of the parameters are allowed to be non-zero are studied in the upcoming paper [7].