  In this paper we study a subfamily of a classic lattice path, the \emph{Dyck
paths}, called \emph{restricted $d$-Dyck} paths, in short $d$-Dyck. A valley of
a Dyck path $P$ is a local minimum of $P$; if the difference between the
heights of two consecutive valleys (from left to right) is at least $d$, we say
that $P$ is a restricted $d$-Dyck path. The \emph{area} of a Dyck path is the
sum of the absolute values of $y$-components of all points in the path. We find
the number of peaks and the area of all paths of a given length in the set of
$d$-Dyck paths. We give a bivariate generating function to count the number of
the $d$-Dyck paths with respect to the the semi-length and number of peaks.
After that, we analyze in detail the case $d=-1$. Among other things, we give
both, the generating function and a recursive relation for the total area.

Combinatorics