### Abstract

Language | English |
---|---|

Number of pages | 41 |

Journal | Integers: Electronic Journal of Combinatorial Number Theory |

Publication status | Accepted/In press - 30 Jul 2015 |

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### Keywords

- mesh patterns
- quadrant marked mesh patterns
- recurrence relations

### Cite this

*Integers: Electronic Journal of Combinatorial Number Theory*.

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*Integers: Electronic Journal of Combinatorial Number Theory*.

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

Research output: Contribution to journal › Article

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 -