Date of Award
8-2025
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematical Sciences
Committee Chair/Advisor
Svetlana Poznanović
Committee Member
Michael Burr
Committee Member
Neil Calkin
Committee Member
Matthew Macauley
Abstract
We study the homomesy phenomenon under the rowmotion operator acting on order ideals of posets. We provide details for the extensions of several results in the literature concerning homomesies of toggleability statistics from finite to infinite orbits, which allows us to obtain homomesies for piecewise-linear and birational rowmotion, even in the case of infinite orbits. Integral to this is a novel generalization of a result in the literature, which relaxes the conditions needed to lift a statistic.
We completely describe the order ideal (resp. antichain) toggleability space for general fences: the space of statistics which are linear combinations of order ideal (antichain) indicator functions and equal to a constant plus a linear combination of toggleability statistics. This allows us to strengthen some homomesies under rowmotion on fences proven by Elizalde et al. and prove some new homomesy results for combinatorial, piecewise-linear, and birational rowmotion, as well as their rank-permuted versions.
Additionally, we completely describe both toggleability spaces for posets whose ideals correspond to Motzkin paths, yielding new homomesy results for combinatorial, piecewise-linear, and birational rowmotion, as well as their rank-permuted versions. We prove other homomesy and orbit structure results by interpreting rowmotion as an action on sequences called whirling.
Recommended Citation
Mertin, Alec, "Homomesies and Toggleability Spaces" (2025). All Dissertations. 4090.
https://open.clemson.edu/all_dissertations/4090
Author ORCID Identifier
0009-0009-3167-7501