# Distributions of several infinite families of mesh patterns

## TL;DR

This paper generalizes known distribution and avoidance results for mesh patterns in permutations, providing formulas for infinite families and extending understanding of their combinatorial properties.

## Contribution

It offers broad generalizations for distribution and avoidance formulas of mesh patterns, including new formulas for infinite families and a new length-2 pattern.

## Key findings

- Generalized 8 distribution results for mesh patterns
- Extended 5 avoidance results to infinite families
- Derived distribution of an additional length-2 mesh pattern

## Abstract

Br\"and\'en and Claesson introduced mesh patterns to provide explicit expansions for certain permutation statistics as linear combinations of (classical) permutation patterns. The first systematic study of avoidance of mesh patterns was conducted by Hilmarsson et al., while the first systematic study of the distribution of mesh patterns was conducted by the first two authors. In this paper, we provide far-reaching generalizations for 8 known distribution results and 5 known avoidance results related to mesh patterns by giving distribution or avoidance formulas for certain infinite families of mesh patterns in terms of distribution or avoidance formulas for smaller patterns. Moreover, as a corollary to a general result, we find the distribution of one more mesh pattern of length 2.