Carlos M. Nicolás

Another bijection between $2$triangulations and pairs of noncrossing Dyck paths
dmtcs:2683 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2009,
DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)

https://doi.org/10.46298/dmtcs.2683
Another bijection between $2$triangulations and pairs of noncrossing Dyck pathsArticle
Authors: Carlos M. Nicolás ^{1}
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Carlos M. Nicolás
1 Department of Mathematics and Statistics
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 $k$ noncrossing Dyck paths of semilength $n2k$. 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.