  A fence is a poset with elements F = {x_1, x_2, ..., x_n} and covers x_1 <
x_2 < ... < x_a > x_{a+1} > ... > x_b < x_{b+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 orbomesy, 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.
