Recurrence solution of monomer-polymer models on two-dimensional rectangular lattices
Yong Kong
TL;DR
The paper addresses counting monomer-polymer coverings (the placement of $k$-mers) on two-dimensional rectangular lattices with monomers occupying the remaining sites, generalizing the classical monomer-dimer problem ($k=2$). It introduces a state-based, row-restriction framework and proves general recurrence relations for the number of configurations with a fixed number of $k$-mers, valid for arbitrary $k$ and lattice width $n$, including explicit boundary-condition dependent constants $c(n,k)$. The central result is the recurrence $\sum_{i=0}^{s} (-1)^i \binom{s}{i} a_{m-i, s} = c(n, k)^s$ for $m \ge ks$, plus a zero-recurrence for $m \ge ks+1$, with a detailed proof employing restricted-row counts and combinatorial identities. The conclusions discuss generating functions and potential implications for the broader #P-complete monomer-dimer counting problem, highlighting the method's potential for insights into longstanding complexity questions.
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.