Recurrence solution of monomer-polymer models on two-dimensional rectangular lattices

TL;DR

This paper derives recurrence relations for counting polymer arrangements on 2D rectangular lattices, generalizing the monomer-dimer problem and potentially aiding in solving complex enumeration problems.

Contribution

It introduces general recurrence relations for monomer-polymer models on 2D lattices, extending previous specific cases like the monomer-dimer problem.

Findings

Recurrence relations applicable for arbitrary polymer length and lattice width.

The relations generalize the monomer-dimer problem.

Potential implications for solving #P-complete enumeration problems.

Abstract

The problem of counting polymer coverings on the rectangular lattices is investigated. In this model, a linear rigid polymer covers $k$ adjacent lattice sites such that no two polymers occupy a common site. Those unoccupied lattice sites are considered as monomers. We prove that for a given number of polymers ($k$-mers), the number of arrangements for the polymers on two-dimensional rectangular lattices satisfies simple recurrence relations. These recurrence relations are quite general and apply for arbitrary polymer length ($k$) and the width of the lattices ($n$). The well-studied monomer-dimer problem is a special case of the monomer-polymer model when $k=2$. It is known the enumeration of monomer-dimer configurations in planar lattices is #P-complete. The recurrence relations shown here have the potential for hints for the solution of long-standing problems in this class of computational complexity.