Séminaire Lotharingien de Combinatoire, 86B.40 (2022), 12 pp.
Sergi Elizalde, Matthew Plante, Tom Roby and Bruce Sagan
Rowmotion on Fences
Abstract.
A fence is a poset with elements F={x1,x2,...,xn} and covers
x1 < x2 < ... < xa > xa+1 > ... > xb < xb+1 < ...,
where a,b,... are positive integers. We investigate rowmotion on antichains and ideals
of F. In particular, we show that orbits of antichains can be visualized using tilings.
This permits us to prove various homomesy results for the number of elements of an antichain
or ideal in an orbit. Rowmotion on fences also exhibits a new phenomenon, which we call
homometry, where the value of a statistic is constant on orbits of the same size. Along the
way, we prove a homomesy result for all self-dual posets and show that any two Coxeter
elements in certain toggle groups behave similarly with respect to homomesies which are
linear combinations of ideal indicator functions. We end with some conjectures and avenues
for future research.
Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.
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