Quadrant marked mesh patterns in 132-avoiding permutations I

TL;DR

This paper investigates the distribution of quadrant marked mesh patterns in 132-avoiding permutations, focusing on cases where only one parameter is non-zero, extending previous systematic studies in permutation pattern analysis.

Contribution

It provides a detailed analysis of the distribution of specific quadrant marked mesh patterns in 132-avoiding permutations, focusing on single-parameter cases, and sets the stage for future multi-parameter studies.

Findings

Distribution formulas for MMP(a,b,c,d) with one non-zero parameter

Characterization of pattern occurrences in 132-avoiding permutations

Foundation for analyzing more complex pattern distributions

Abstract

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.