Quadrant marked mesh patterns in 132-avoiding permutations I

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.