On fourteen equidistribution conjectures of Lv and Zhang and monotone mesh patterns with corner shadings

TL;DR

This paper proves and generalizes fourteen equidistribution conjectures related to permutations using involutions, and provides enumerative results for pattern-avoiding permutations with specific mesh pattern restrictions.

Contribution

It introduces three involutions that prove and extend all remaining conjectures of Lv and Zhang, and offers new enumeration results for restricted mesh pattern-avoiding permutations.

Findings

Proved all fourteen conjectures of Lv and Zhang.

Generalized conjectures using involutions.

Enumerated classes of pattern-avoiding permutations.

Abstract

Three complementation-like involutions are constructed on permutations to prove, and in some cases generalize, all remaining fourteen joint symmetric equidistribution conjectures of Lv and Zhang. Further enumerative results are obtained for several classes of (mesh) pattern-avoiding permutations, where the shadings of all involved mesh patterns are restricted to an opposing pair of corners.