An Infinite Family of Adsorption Models and Restricted Lukasiewicz Paths

We define (k,?)-restricted Lukasiewicz paths, k??????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,?). The resulting polynomial is of degree ?+1 and hence cannot be solved explicitly for sufficiently large ?. We provide two different approaches to obtain the phase diagram. In addition to a more conventional analysis, we also develop a new mathematical characterisation 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,?)-restricted Lukasiewicz paths and ??rise??-restricted Dyck paths, identifying another family of path models which share the same critical behaviour. For (k,?)=(1,??) 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,?)-restricted Lukasiewicz paths, and for (k,?)=(0,??) we provide a bijection to Dyck paths.