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Abstract: Triangulations of a convex polygon are known to be counted by the Catalan numbers. A natural generalization of a triangulation is a k-triangulation, which is defined to be a maximal set of diagonals so that no k+1 of them mutually cross in their interiors. It was proved by Jakob Jonsson that k-triangulations are enumerated by certain determinants of Catalan numbers, that are also known to count k-tuples of non-crossing Dyck paths.
There are several simple bijections between triangulations of a convex n-gon and Dyck paths. However, no bijective proof of Jonsson's result is known for general k. In this talk I will give a bijective proof for the case k=2, that is, I will present a bijection between 2-triangulations of a convex n-gon and pairs (P,Q) of Dyck paths of semilength n-4 so that P never goes below Q. The bijection is obtained by constructing isomorphic generating trees for the sets of 2-triangulations and pairs of non-crossing Dyck paths.
This talk will be accessible to undergraduates.
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