Quadrant marked mesh patterns in 132-avoiding permutations III

Sergey Kitaev, Jeffrey Remmel, Mark Tiefenbruck

Research output: Contribution to journalArticle

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 132-avoiding permutations started in [9] and [10] where we studied the distribution of the number of matches of MMP(a, b, c, d) in 132-avoiding 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 132-avoiding 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.
LanguageEnglish
Number of pages41
JournalIntegers: Electronic Journal of Combinatorial Number Theory
Publication statusAccepted/In press - 30 Jul 2015

Fingerprint

Quadrant
Permutation
Mesh
Generating Function
Zero
Recurrence relation
Symmetric group
Continuation
Closed-form
Coefficient

Keywords

  • mesh patterns
  • quadrant marked mesh patterns
  • recurrence relations

Cite this

@article{3294b181af79497987a31b6cf1c33e05,
title = "Quadrant marked mesh patterns in 132-avoiding permutations III",
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 132-avoiding permutations started in [9] and [10] where we studied the distribution of the number of matches of MMP(a, b, c, d) in 132-avoiding 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 132-avoiding 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.",
keywords = "mesh patterns, quadrant marked mesh patterns, recurrence relations",
author = "Sergey Kitaev and Jeffrey Remmel and Mark Tiefenbruck",
year = "2015",
month = "7",
day = "30",
language = "English",
journal = "Integers: Electronic Journal of Combinatorial Number Theory",
issn = "1553-1732",

}

Quadrant marked mesh patterns in 132-avoiding permutations III. / Kitaev, Sergey; Remmel, Jeffrey; Tiefenbruck, Mark.

In: Integers: Electronic Journal of Combinatorial Number Theory, 30.07.2015.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Quadrant marked mesh patterns in 132-avoiding permutations III

AU - Kitaev, Sergey

AU - Remmel, Jeffrey

AU - Tiefenbruck, Mark

PY - 2015/7/30

Y1 - 2015/7/30

N2 - 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 132-avoiding permutations started in [9] and [10] where we studied the distribution of the number of matches of MMP(a, b, c, d) in 132-avoiding 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 132-avoiding 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.

AB - 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 132-avoiding permutations started in [9] and [10] where we studied the distribution of the number of matches of MMP(a, b, c, d) in 132-avoiding 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 132-avoiding 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.

KW - mesh patterns

KW - quadrant marked mesh patterns

KW - recurrence relations

UR - http://www.integers-ejcnt.org/

M3 - Article

JO - Integers: Electronic Journal of Combinatorial Number Theory

T2 - Integers: Electronic Journal of Combinatorial Number Theory

JF - Integers: Electronic Journal of Combinatorial Number Theory

SN - 1553-1732

ER -