  The aim of this paper is to develop the combinatorics of constructions
associated to what we call \emph{triangular partitions}. As introduced in
arXiv:2102.07931, these are the partitions whose cells are those lying below
the line joining points $(r,0)$ and $(0,s)$, for any given positive reals $r$
and $s$. Classical notions such as Dyck paths and parking functions are
naturally generalized by considering the set of partitions included in a given
triangular partition. One of our striking results is that the restriction of
the Young lattice to triangular partition has a planar Hasse diagram, with many
nice properties. It follows that we may generalize the "first-return"
recurrence, for the enumeration of classical Dyck paths, to the enumeration of
all partitions contained in a fixed triangular one.
