  A \emph{Dyck path} is a lattice path in the first quadrant of the $xy$-plane
that starts at the origin, ends on the $x$-axis, and consists of the same
number of North-East steps $U$ and South-East steps $D$. A \emph{valley} is a
subpath of the form $DU$. A Dyck path is called \emph{restricted $d$-Dyck} if
the difference between any two consecutive valleys is at least $d$ (right-hand
side minus left-hand side) or if it has at most one valley. In this paper we
give some connections between restricted $d$-Dyck paths and both, the
non-crossing partitions of $[n]$ and some subfamilies of polyominoes. We also
give generating functions to count several aspects of these combinatorial
objects.
