  Let Q be a Dynkin quiver of type A. The bounded derived category of the path
algebra of Q has an autoequivalence given by the composition of the
Auslander-Reiten translate and the square of the shift functor. We classify the
maximal rigid objects in the corresponding orbit category C(Q), in terms of
bipartite noncrossing graphs (with loops) in a circle. We also describe the
endomorphism algebras of the maximal rigid objects, and we prove that a certain
class of these algebras are iterated tilted algebras of type A.
