Quadrant marked mesh patterns in 132-avoiding permutations I

This paper is a continuation of the systematic study of the distributions of quadrant marked mesh patterns initiated in [6]. Given a permutation \(\sg = \sg_1 ... \sg_n\) in the symmetric group \(S_n\), we say that \(\sg_i\) matches the quadrant marked mesh pattern \(MMP(a,b,c,d)\) if there are at l...

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Bibliographic Details
Published in:arXiv.org 2014-07
Main Authors: Kitaev, Sergey, Remmel, Jeffrey, Tiefenbruck, Mark
Format: Article
Language:English
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Parameters
Permutations
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Summary:This paper is a continuation of the systematic study of the distributions of quadrant marked mesh patterns initiated in [6]. Given a permutation \(\sg = \sg_1 ... \sg_n\) in the symmetric group \(S_n\), we say that \(\sg_i\) matches the quadrant marked mesh pattern \(MMP(a,b,c,d)\) if there are at least \(a\) elements to the right of \(\sg_i\) in \(\sg\) that are greater than \(\sg_i\), at least \(b\) elements to left of \(\sg_i\) in \(\sg\) that are greater than \(\sg_i\), at least \(c\) elements to left of \(\sg_i\) in \(\sg\) that are less than \(\sg_i\), and at least \(d\) elements to the right of \(\sg_i\) in \(\sg\) that are less than \(\sg_i\). We study the distribution of \(MMP(a,b,c,d)\) in 132-avoiding permutations. In particular, we study the distribution of \(MMP(a,b,c,d)\), where only one of the parameters \(a,b,c,d\) are non-zero. In a subsequent paper [7], we will study the the distribution of \(MMP(a,b,c,d)\) in 132-avoiding permutations where at least two of the parameters \(a,b,c,d\) are non-zero.
ISSN:2331-8422