  This paper presents a geometric model of the Auslander-Reiten quiver of a
type A quiver together with a stability function for which all indecomposable
modules are stable. We also introduce a new Catalan object which we call a
maximal almost rigid representation. We show that its endomorphism algebra is a
tilted algebra of type A. We define a partial order on the set of maximal
almost rigid representations and use our new geometric model to show that this
partial order is a Cambrian lattice.
