Abstract

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A Ptolemy diagram is a simple combinatorial object: A collection of diagonals 

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in a polygon with the property that if the collection contains two crossing 

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diagonals, then it also contains all other diagonals connecting end points of 

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the crossing ones.  They could have been defined at any point since antiquity 

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but were in fact introduced recently, in an infinite version, by Puiman Ng.  

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We give a new combinatorial description of (finite) Ptolemy diagrams, show how 

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they parametrise torsion pairs in cluster categories of Dynkin type A, and 

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count the number of such torsion pairs.