Considering the path algebra of the quiver $\mathbb{A}_n$, it is well known its Auslander-Reiten quiver with the canonical orientation of $\mathbb{A}_n$, that is, with all the arrows from, say, left to right. I can find as examples in several texts the AR-quivers of $\mathbb{A}_n$ with other orientations.

QUESTION: Is there an algorithm to construct the AR-quiver of $\textit{any}$ orientation of $\mathbb{A}_n$ ?

Clearly it will suffice to describe the effect in the AR-quiver if it is changed the orientation of one arrow. Observing the examples I think there is some pattern, but I can't figure out the algorithm.

Of course this algorithm could be applied to quivers of other type than $\mathbb{A}_n$, but I think it is better to understand this in the simplest case.

knitting procedure(see the book by Gabriel and Roiter for the problem of who to attribute it to...) will construct the AR-quiver of those algebras in a breeze. This is explained in the book by Gabriel and Roiter, in Assem+Skowroński+Simpson, in ARS iirc, and surely elsewhere. $\endgroup$ – Mariano Suárez-Álvarez Jan 29 '12 at 3:29