# Quadrant marked mesh patterns in 132-avoiding permutations II

Kitaev, Sergey and Remmel, Jeffrey and Tiefenbruck, Mark
(2015)
*Quadrant marked mesh patterns in 132-avoiding permutations II.*
Integers: Electronic Journal of Combinatorial Number Theory, 15.
A16.
ISSN 1553-1732

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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 quadrant marked mesh patterns in 132-avoiding permutations started by the present authors where we mainly studied the distribution of the number of matches of MMP(a,b,c,d) in 132-avoiding permutations where exactly one of a,b,c,d is 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 132-avoiding permutations where exactly two of a,b,c,d are greater than zero and the remaining elements are 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. The case of quadrant marked mesh patterns MMP(a,b,c,d) where three or more of a,b,c,d are constrained to be greater than 0 will be studied in a future article by the present authors.

#### ORCID iDs

Kitaev, Sergey ORCID: https://orcid.org/0000-0003-3324-1647, Remmel, Jeffrey and Tiefenbruck, Mark;Item type: Article ID code: 53760 Dates: DateEvent8 May 2015Published21 April 2015AcceptedKeywords: mesh patterns, enumeration, permutation statistics, Electronic computers. Computer science, Computational Theory and Mathematics Subjects: Science > Mathematics > Electronic computers. Computer science Department: Faculty of Science > Computer and Information Sciences Depositing user: Pure Administrator Date deposited: 14 Jul 2015 14:39 Last modified: 14 Apr 2021 00:31 Related URLs: URI: https://strathprints.strath.ac.uk/id/eprint/53760