Abstract:
We derive a simple bijection between geometric plane perfect matchings on $2n$ points in convex position and triangulations on $n+2$ points in convex position. We then extend this bijection to monochromatic plane perfect matchings on periodically $k$-colored vertices and $(k+2)$-gonal tilings of convex point sets. These structures are related to a generalization of Temperley-Lieb algebras and our bijections provide explicit one-to-one relations between matchings and tilings.
In the second part of the talk we focus on combinatorial aspects and applications of this bijection.
This is joint work with Oswin Aichholzer, Karin Baur and Birgit Vogtenhuber.