Perfect k-Colored Matchings and (k+2)-Gonal Tilings

Oswin Aichholzer,Birgit Vogtenhuber,Lukas Andritsch,Karin Baur

doi:10.1007/s00373-018-1967-8

Oswin Aichholzer, Birgit Vogtenhuber + Show 2 more

Open Access

https://doi.org/10.1007/s00373-018-1967-8

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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. Moreover, for a given element of one class, the corresponding element of the other class can be computed in linear time.

Highlights

Introduction1 km+m m m−1 are known to count the number of (k + 2)-gonal tilings of a convex polygon of size km + 2 and go back to Fuss-
The Fuss–Catalan numbers f (k, m) =1 km+m m m−1 are known to count the number of (k + 2)-gonal tilings of a convex polygon of size km + 2 and go back to Fuss-Euler
Since the number of (k + 2)-gonal tilings coincides with the number of k-colored matchings, these sets are in bijection

Summary

Introduction

1 km+m m m−1 are known to count the number of (k + 2)-gonal tilings of a convex polygon of size km + 2 and go back to Fuss-. Bisch and Jones introduced k-colored Fuss–Catalan algebras in [1] as a natural generalization of Temperley–Lieb algebras. These algebras have bases by certain planar k-colored diagrams with mk vertices on top and bottom. Graphs and Combinatorics (2018) 34:1333–1346 of such an algebra is f (k, m), with a basis indexed by these diagrams. Since the number of (k + 2)-gonal tilings coincides with the number of k-colored matchings, these sets are in bijection. Our main theorems are the explicit bijections between the set of k-colored matchings and (k + 2)-gonal tilings (Theorems 1 and 3). A key ingredient is the characterization of valid k-colored matchings in Theorem 2

Temperley–Lieb Algebras

Fuss–Catalan Algebras

Matchings and Triangulations

Matchings with k Colors

Tilings with t-Gons

Future Work