Joint equidistributions of mesh patterns 123 and 321 with symmetric and minus-antipodal shadings

TL;DR

This paper proves extensive joint equidistribution results for mesh patterns 123 and 321 with specific shadings, linking these combinatorial structures to well-known integer sequences and extending prior work in the field.

Contribution

It significantly broadens the understanding of joint equidistributions of mesh patterns with symmetric and minus-antipodal shadings, covering most cases and connecting to classical sequences.

Findings

Proved 20 out of 22 joint equidistributions with symmetric shadings.

Established all 36 joint equidistributions with minus-antipodal shadings.

Linked these distributions to Stirling numbers, harmonic numbers, and inversion sequence counts.

Abstract

In this paper, we extend recent results by Lv and Kitaev by proving 20 (out of 22 possible) joint equidistributions of mesh patterns 123 and 321 with symmetric shadings, as well as all 36 joint equidistributions of these patterns with minus-antipodal shadings. Our results link several joint equidistributions of mesh patterns to various integer sequences, including unsigned Stirling numbers of the first kind, harmonic numbers, and the numbers of inversion sequences avoiding a certain vincular pattern studied by Lin and Yan.