  We define $(k,\ell)$-restricted Lukasiewicz paths, $k\le\ell\in\mathbb{N}_0$,
and use these paths as models of polymer adsorption. We write down a polynomial
expression satisfied by the generating function for arbitrary values of
$(k,\ell)$. The resulting polynomial is of degree $\ell+1$ and hence cannot be
solved explicitly for sufficiently large $\ell$. We provide two different
approaches to obtain the phase diagram. In addition to a more conventional
analysis, we also develop a new mathematical characterization of the phase
diagram in terms of the discriminant of the polynomial and a zero of its
highest degree coefficient.
  We then give a bijection between $(k,\ell)$-restricted Lukasiewicz paths and
"rise"-restricted Dyck paths, identifying another family of path models which
share the same critical behaviour. For $(k,\ell)=(1,\infty)$ we provide a new
bijection to Motzkin paths.
  We also consider the area-weighted generating function and show that it is a
q-deformed algebraic function. We determine the generating function explicitly
in particular cases of $(k,\ell)$-restricted Lukasiewicz paths, and for
$(k,\ell)=(0,\infty)$ we provide a bijection to Dyck paths.
