Abstract

A $k$-triangulation of the $n$-gon is a maximal set of diagonals of the $n$-gon containing no subset of $k+1$ mutually crossing diagonals. The number of $k$-triangulations of the $n$-gon, determined by Jakob Jonsson, is equal to a $k \times k$ Hankel determinant of Catalan numbers. This determinant is also equal to the number of $k$ non-crossing Dyck paths of semi-length $n-2k$. This brings up the problem of finding a combinatorial bijection between these two sets. In FPSAC 2007, Elizalde presented such a bijection for the case $k=2$. We construct another bijection for this case that is stronger and simpler that Elizalde's. The bijection preserves two sets of parameters, degrees and generalized returns. As a corollary, we generalize Jonsson's formula for $k=2$ by counting the number of $2$-triangulations of the $n$-gon with a given degree at a fixed vertex. Une $k$-triangulation du $n$-gon est un ensemble maximal de diagonales du $n$-gon ne contenant pas de sous-ensemble de $k+1$ diagonales mutuellement croisant. Le nombre de $k$-triangulations du $n$-gon, déterminé par Jakob Jonsson, est égal à un déterminant de Hankel $k \times k$ de nombres de Catalan. Ce déterminant est aussi égal au nombre de $k$ chemins de Dyck de largo $n-2k$ que ne pas se croiser. Cela porte le problème de trouver une bijection de type combinatoire entre ces deux ensembles. À la FPSAC 2007, Elizalde a présenté une telle bijection pour le cas $k = 2$. Nous construisons une autre bijection pour ce cas qui est plus forte et plus simple que de l'Elizalde. La bijection conserve deux ensembles de paramètres, les degré et les retours généralisée. De ce, nous généralisons la formule de Jonsson pour $k = 2$ en comptant le nombre de $2$-triangulations du $n$-gon avec un degré à un vertex fixe.

Highlights

  • The set of triangulations of n points in convex position on the plane has been studied for a long time because of its interesting combinatorial properties

  • A combinatorial bijection between the set of k-triangulations of the n-gon and the corresponding set of k non-crossing Dyck paths would constitute a simpler proof of the formula for the number of k-triangulations

  • By means of a family of involutions, that the distribution of non-crossing Dyck paths with respect to these generalized returns is independent of their order

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Summary

Introduction

The set of triangulations of n points in convex position on the plane has been studied for a long time because of its interesting combinatorial properties. A combinatorial bijection between the set of k-triangulations of the n-gon and the corresponding set of k non-crossing Dyck paths would constitute a simpler proof of the formula for the number of k-triangulations. Our bijection is stronger than that in [6] because it transforms two simple parameters for 2-triangulations, the degrees (number of neighbors) at two consecutive vertices, into two simple parameters for pairs of non-crossing Dyck paths, the number of (generalized) returns. Dyck paths have the same distribution with respect to the height of the first peak and with respect to the number of returns This result is a particular case of a theorem originally found by Brak and Essam [2]. The bijection sends the 2-triangulations having degrees c0 and c1 at two fixed consecutive vertices onto the set of all pairs of non-crossing Dyck paths having (generalized) returns c0 and c1. We conjecture that a similar formula holds for every k

The set of k-triangulations
The set of non-crossing Dyck paths
A strong bijection between Tn2 and Dn2
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