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.Comment: Theorem 10 is correcte