Quadrant marked mesh patterns in 132avoiding permutations III
Kitaev, Sergey and Remmel, Jeffrey and Tiefenbruck, Mark (2015) Quadrant marked mesh patterns in 132avoiding permutations III. Integers: Electronic Journal of Combinatorial Number Theory. ISSN 15531732 (In Press)

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Abstract
Given a permutation σ = σ1 . . . σn in the symmetric group Sn, we say that σi matches the marked mesh pattern MMP(a, b, c, d) in σ if there are at least a points to the right of σi in σ which are greater than σi, at least b points to the left of σi in σ which are greater than σi, at least c points to the left of σi in σ which are smaller than σi, and at least d points to the right of σi in σ which are smaller than σi. This paper is continuation of the systematic study of the distributions of quad rant marked mesh patterns in 132avoiding permutations started in [9] and [10] where we studied the distribution of the number of matches of MMP(a, b, c, d) in 132avoiding permutations where at most two elements of a, b, c, d are greater than zero and the remaining elements are zero. In this paper, we study the distribution of the number of matches of MMP(a, b, c, d) in 132avoiding permutations where at least three of a, b, c, d are greater than zero. We provide explicit recurrence relations to enumerate our objects which can be used to give closed forms for the generating functions associated with such distributions. In many cases, we provide combinatorial explanations of the coefficients that appear in our generating functions.
Item type:  Article 

ID code:  54142 
Keywords:  mesh patterns, quadrant marked mesh patterns, recurrence relations, Mathematics, Discrete Mathematics and Combinatorics 
Subjects:  Science > Mathematics 
Department:  Faculty of Science > Computer and Information Sciences 
Depositing user:  Pure Administrator 
Date deposited:  02 Sep 2015 15:45 
Last modified:  13 Feb 2019 13:11 
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URI:  https://strathprints.strath.ac.uk/id/eprint/54142 
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