Sort-Invariant Non-Messing-Up

TL;DR

This paper characterizes labelings of convex subposets of a cylinder that remain invariant under a specific two-step sorting process, linking this invariance to avoiding a particular diamond pattern.

Contribution

It provides a complete characterization of sort-invariant labelings for convex subposets of a cylinder based on pattern avoidance.

Findings

Sort-invariant labelings are characterized by avoiding a specific diamond pattern.

The characterization applies to convex subposets of a cylinder.

The property ensures the order is independent of sorting sequence.

Abstract

A poset has the non-messing-up property if it has two covering sets of disjoint saturated chains so that for any labeling of the poset, sorting the labels along one set of chains and then sorting the labels along the other set yields a linear extension of the poset. The linear extension yielded by thus twice sorting a labeled non-messing-up poset may be independent of which sort was performed first. Here we characterize such sort-invariant labelings for convex subposets of a cylinder. They are completely determined by avoidance of a particular subpattern: a diamond of four elements whose smallest two labels appear at opposite points.