Joint equidistributions of mesh patterns 123 and 132 with minus antipodal shadings

TL;DR

This paper advances the understanding of joint equidistributions of specific mesh patterns by proving numerous new cases with minus antipodal shadings, using bijections, recurrences, and generating functions, and linking results to Stirling numbers.

Contribution

It significantly extends previous work by proving 112 new joint equidistributions of mesh patterns with minus antipodal shadings and introduces methods connecting these distributions to Stirling numbers.

Findings

Proved 112 new joint equidistributions with minus antipodal shadings.

Constructed bijections, recurrence relations, and generating functions for these distributions.

Linked joint distributions of mesh patterns to unsigned Stirling numbers of the first kind.

Abstract

The study of joint equidistributions of mesh patterns 123 and 132 with the same symmetric shadings was recently initiated by Kitaev and Lv, where 75 of 80 potential joint equidistributions were proven. In this paper, we prove 112 out of 126 potential joint equidistributions of mesh patterns 123 and 132 with the same minus antipodal shadings. As a byproduct, we present 562 joint equidistribution results for non-symmetric and non-minus-antipodal shadings. To achieve this, we construct bijections, find recurrence relations, and obtain generating functions. Moreover, we demonstrate that the joint distributions of several pairs of mesh patterns are related to the unsigned Stirling numbers of the first kind.