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Séminaire Lotharingien de Combinatoire, B88a (2023), 21 pp.

# Rowmotion on Rooted Trees

# Pranjal Dangwal, Jamie Kimble, Jinting Liang, Jianzhi Lou, Bruce E. Sagan
and Zach Stewart

**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 has a
constant sum 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.

Received: August 25, 2022.
Revised: April 5, 2023.
Accepted: May 16, 2023.

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