The Catalan numbers are one of the most ubiquitous and fascinating sequences of enumerative combinatorics (Stanley), in particular they count the number of non-crossing partitions of a finite set. In the appendix of these notes we will try to outline in which way the Catalan combinatorics could be seen as the heart of the theory of finite sets, starting with the subsets of cardinality two. If we fix a finite set C of cardinality n + 1, the subsets of cardinality two may be considered as the positive roots of a root system (in the sense of Lie theory) of Dynkin type A_n. There are recent proposals to work with generalized non-crossing partitions, starting with any root system (of Dynkin type A_n, B_n,..., G2). The Catalan combinatorics looks for sets of partitions of C which are of relevance and relates them to subsets of the automorphism group S_n+1= Aut(C), this is the Weyl group of type A. The generalized Cartan combinatorics starts directly with a suitable subset of G, where G is any Weyl (or, more generally, any Coxeter) group. It turns out that the representation theory of representation-finite hereditary artin algebras A can be used in order to categorify these generalized non-crossing partitions in the Weyl group case. In particular, for the case A_n, one may use the ring A_n of all upper triangular (n × n)-matrices with coefficients in a field.
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