An infinite family of adsorption models and restricted Lukasiewicz paths

TL;DR

This paper introduces a new class of Lukasiewicz paths as models for polymer adsorption, develops polynomial equations for their generating functions, and explores their phase diagrams and bijections to other path models.

Contribution

It defines $(k,\, ext{ell})$-restricted Lukasiewicz paths, derives their generating functions, and establishes bijections with Dyck and Motzkin paths, revealing new insights into their critical behavior.

Findings

Polynomial generating functions of degree $ ext{ell}+1$

New phase diagram characterization via polynomial discriminant

Explicit bijections to Dyck and Motzkin paths

Abstract

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.