Quadrant marked mesh patterns in 132-avoiding permutations III

Given a permutation $\sg = \sg_1 \ldots \sg_n$ in the symmetric group $S_n$, we say that $\sg_i$ matches the marked mesh pattern $MMP(a,b,c,d)$ in $\sg$ if there are at least $a$ points to the right of $\sg_i$ in $\sg$ which are greater than $\sg_i$, at least $b$ points to the left of $\sg_i$ in $\sg$ which are greater than $\sg_i$, at least $c$ points to the left of $\sg_i$ in $\sg$ which are smaller than $\sg_i$, and at least $d$ points to the right of $\sg_i$ in $\sg$ which are smaller than $\sg_i$. This paper is continuation of the systematic study of the distribution of quadrant marked mesh patterns in 132-avoiding permutations started in \cite{kitremtie} and \cite{kitremtieII} 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 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.

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