# Quadrant marked mesh patterns in 123-avoiding permutations

## TL;DR

This paper investigates the distribution of quadrant marked mesh pattern matches in 123-avoiding permutations, extending previous work on other pattern-avoiding classes, and provides recurrence relations and combinatorial insights.

## Contribution

It introduces explicit recurrence relations and combinatorial explanations for the distribution of mesh pattern matches in 123-avoiding permutations, a new class not previously analyzed.

## Key findings

- Derived recurrence relations for counting pattern matches.
- Provided closed-form generating functions for distributions.
- Offered combinatorial explanations for coefficients in generating functions.

## Abstract

Given a permutation $\sigma = \sigma_1 \ldots \sigma_n$ in the symmetric group $\mathcal{S}_{n}$, we say that $\sigma_i$ matches the quadrant marked mesh pattern $\mathrm{MMP}(a,b,c,d)$ in $\sigma$ if there are at least $a$ points to the right of $\sigma_i$ in $\sigma$ which are greater than $\sigma_i$, at least $b$ points to the left of $\sigma_i$ in $\sigma$ which are greater than $\sigma_i$, at least $c$ points to the left of $\sigma_i$ in $\sigma$ which are smaller than $\sigma_i$, and at least $d$ points to the right of $\sigma_i$ in $\sigma$ which are smaller than $\sigma_i$. 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.