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

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.