Adsorbing Motzkin paths

The adsorption of a Motzkin path with lifted endpoints onto an adsorbing line is examined as a model of polymer adsorption. The partition function of this model is determined exactly, and the free energy is derived by examining the partition function. The location of the critical point in the model is given exactly in one case, and as the solution of a nonlinear equation in another case. The phase diagram of the model is examined, and I show that the transition is in general a first-order adsorption transition, except in cases where one or both endpoints of the path are restricted to be close to the adsorbing line.