  A new hierarchy of operads over the linear spans of $\delta$-cliffs, which
are some words of integers, is introduced. These operads are intended to be
analogues of the operad of permutations, also known as the associative
symmetric operad. We obtain operads whose partial compositions can be described
in terms of intervals of the lattice of $\delta$-cliffs. These operads are very
peculiar in the world of the combinatorial operads since, despite to the
relative simplicity for their construction, they are infinitely generated and
they have nonquadratic and nonhomogeneous nontrivial relations. We provide a
general construction for some of their quotients. We use it to endow the spaces
of permutations, $m$-increasing trees, $c$-rectangular paths, and $m$-Dyck
paths with operad structures. The operads on $c$-rectangular paths admit, as
Koszul duals, operads generalizing the duplicial and triplicial operads.
