Rowmotion on rooted trees

TL;DR

This paper introduces a tiling model to analyze rowmotion on rooted trees, revealing homometry properties for certain statistics, extending previous work on fences and exploring orbit structures in posets.

Contribution

It develops a new tiling framework for rowmotion on rooted trees and demonstrates homometry phenomena for specific statistics, broadening understanding beyond fences.

Findings

Rooted trees exhibit homometry under rowmotion for certain statistics.

The tiling model effectively describes orbit structures in rooted trees.

Homomesy does not generally hold, but homometry does in studied cases.

Abstract

A rooted tree T is a poset whose Hasse diagram is a graph-theoretic tree having a unique minimal element. We study rowmotion on antichains and lower order ideals of T. Recently Elizalde, Roby, Plante and Sagan considered rowmotion on fences which are posets whose Hasse diagram is a path (but permitting any number of minimal elements). They showed that in this case, the orbits could be described in terms of tilings of a cylinder. They also defined a new notion called homometry which means that a statistic takes a constant value on all orbits of the same size. This is a weaker condition than the well-studied concept of homomesy which requires a constant value for the average of the statistic over all orbits. Rowmotion on fences is often homometric for certain statistics, but not homomesic. We introduce a tiling model for rowmotion on rooted trees. We use it to study various specific types of trees and show that they exhibit homometry, although not homomesy, for certain statistics.