Summary: The Dynkin algebras are the hereditary artin algebras of finite representation type. The paper exhibits the number of support-tilting modules for any Dynkin algebra. Since the support-tilting modules for a Dynkin algebra of Dynkin type $\Delta $ correspond bijectively to the generalized non-crossing partitions of type $\Delta $, the calculations presented here may also be considered as a categorification of results concerning the generalized non-crossing partitions. In the Dynkin case A, we obtain the Catalan triangle, in the cases B and C the increasing part of the Pascal triangle, and finally in the case D an expansion of the increasing part of the Lucas triangle.
Dynkin algebra, Dynkin diagram, tilting module, support-tilting module, lattice of non-crossing partitions, cluster combinatorics, generalized Catalan number, Catalan triangle, Pascal triangle, Lucas triangle, categorification