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

Nicolás, Carlos M.

Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) (2009).

Résumé

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.

Détail

DOI : 10.46298/dmtcs.2683