Equidistributions of mesh patterns of length two and Kitaev and Zhang's conjectures

TL;DR

This paper proves new equidistribution results for mesh patterns of length two and confirms four conjectures by Kitaev and Zhang using involutions on permutations.

Contribution

It introduces two involutions on permutations to establish equidistributions and verifies four conjectures related to mesh pattern distributions.

Findings

Confirmed four conjectures by Kitaev and Zhang.

Established new equidistribution results for mesh patterns.

Used involutions to analyze permutation patterns.

Abstract

A systematic study of avoidance of mesh patterns of length 2 was conducted by Hilmarsson et al. in 2015. In a recent paper Kitaev and Zhang examined the distribution of the aforementioned patterns. The aim of this paper is to prove more equidistributions of mesh pattern and confirm Kitaev and Zhang's four conjectures by constructing two involutions on permutations.