A bijection between 2-triangulations and pairs of non-crossing Dyck paths

A k-triangulation of a convex polygon is a maximal set of diagonals so that no k + 1 of them mutually cross in their interiors. We present a bijection between 2-triangulations of a convex n-gon and pairs of non-crossing Dyck paths of length 2(n − 4). This gives a bijective proof of a recent result of Jonsson for the case k = 2. We obtain the bijection by constructing isomorphic generating trees for the sets of 2-triangulations and pairs of non-crossing Dyck paths. R´esum´e. Une k-triangulation d’un polygone convexe est un ensemble maximal de diagonales tel qu’il n’y ai pas k +1 diagonales qui se croisent mutuellement en leurs int´erieurs. Nous pr´esentons une bijection entre les 2-triangulations d’un n-gon convexe et paires de chemins de Dyck de longeur 2(n − 4) qui ne se croisent pas. Ceci donne une preuve bijective d’un r´esultat de Jonsson pour le cas k = 2. Nous obtenons cette bijection en construisant arbres g´en´erateurs isomorphes aux ensembles de 2-triangul...

• Type: Journal
• Year: 2007
• Where: JCT
• Authors: Sergi Elizalde
• Keywords: Isomorphic Generating Trees, JCT 2006, JCT 2007, Non-crossing Dyck Paths, Qui Ne