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Ptolemy diagrams and torsion pairs in the cluster category of Dynkin type $A _{n }$. (English)
J. Algebr. Comb. 34, No. 3, 507-523 (2011).
Summary: 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 {\it P. Ng} [“A characterization of torsion theories in the cluster category of Dynkin type $A _{\infty }$", \url{http://arxiv.org/abs/1005.4364v1}]. 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.