Quadrant marked mesh patterns in 123-avoiding permutations

Abstract

Given a permutation $σ= σ_1 \ldots σ_n$ in the symmetric group $\mathcal{S}_{n}$, we say that $σ_i$ matches the quadrant marked mesh pattern $\mathrm{MMP}(a,b,c,d)$ in $σ$ if there are at least $a$ points to the right of $σ_i$ in $σ$ which are greater than $σ_i$, at least $b$ points to the left of $σ_i$ in $σ$ which are greater than $σ_i$, at least $c$ points to the left of $σ_i$ in $σ$ which are smaller than $σ_i$, and at least $d$ points to the right of $σ_i$ in $σ$ which are smaller than $σ_i$. Kitaev, Remmel, and Tiefenbruck systematically studied the distribution of the number of matches of $\mathrm{MMP}(a,b,c,d)$ in 132-avoiding permutations. The operation of reverse and complement on permutations allow one to translate their results to find the distribution of the number of $\mathrm{MMP}(a,b,c,d)$ matches in 231-avoiding, 213-avoiding, and 312-avoiding permutations. In this paper, we study the distribution of the number of matches of $\mathrm{MMP}(a,b,c,d)$ in 123-avoiding permutations. 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.