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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