Ptolemy diagrams and torsion pairs in the cluster category of Dynkin type A n

Abstract

We give a complete classification of torsion pairs in the cluster category of Dynkin type A n . Along the way we give a new combinatorial description of Ptolemy diagrams, an infinite version of which was introduced by Ng (1005.4364v1 [math.RT], 2010). This allows us to count the number of torsion pairs in the cluster category of type A n . We also count torsion pairs up to Auslander–Reiten translation.