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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