Triangulations of a convex polygon are known to be counted by the Catalan
natural generalization of a triangulation is a $k$-triangulation, which is
be a maximal set of diagonals so that no $k+1$ of them mutually cross in
interiors. It was proved by Jakob Jonsson that $k$-triangulations are
certain determinants of Catalan numbers, that are also known to count
non-crossing Dyck paths.
There are several simple bijections between triangulations of a convex
Dyck paths. However, no bijective proof of Jonsson's result is known for
In this talk I will give a bijective proof for the case $k=2$, that is, I
a bijection between $2$-triangulations of a convex $n$-gon and pairs
$(P,Q)$ of Dyck
paths of semilength $n-4$ so that $P$ never goes below $Q$. The bijection
by constructing isomorphic generating trees for the sets of
2-triangulations and pairs
of non-crossing Dyck paths.