Geodesic
Ptolemy diagrams and torsion pairs in the cluster category of Dynkin type $A _{n }$
Journal of Algebraic Combinatorics, Tome 34 (2011) no. 3, pp. 507-523.

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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 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.
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Keywords: keywords clique, cluster algebra, cluster tilting object, generating function, recursively defined set, species, triangulated category 
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     title = {Ptolemy diagrams and torsion pairs in the cluster category of {Dynkin} type $A _{n }$},
     journal = {Journal of Algebraic Combinatorics},
     pages = {507--523},
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Holm, Thorsten; Jørgensen, Peter; Rubey, Martin. Ptolemy diagrams and torsion pairs in the cluster category of Dynkin type $A _{n }$. Journal of Algebraic Combinatorics, Tome 34 (2011) no. 3, pp. 507-523. http://geodesic.mathdoc.fr/item/JAC_2011__34_3_a1/