Regular structures of an intractable enumeration problem: a diagonal recurrence relation of monomer-polymer coverings on two-dimensional rectangular lattices

TL;DR

This paper derives a simple recurrence relation for counting polymer coverings on 2D lattices, advancing understanding of an otherwise intractable enumeration problem in statistical mechanics.

Contribution

It establishes a diagonal recurrence relation for the number of monomer-polymer coverings on rectangular lattices, providing a new analytical tool for this complex counting problem.

Findings

Derived a recurrence relation for polymer coverings

Connected the problem to a known combinatorial formula

Enhanced analytical understanding of intractable enumeration

Abstract

In the monomer-polymer model, a linear rigid polymer covers $k$ adjacent lattice sites, with no lattice site occupied by more than one polymer. The polymers are called $k$-mers, and those unoccupied lattice sites are called monomers. The well-known monomer-dimer model is a special case of the monomer-polymer model with $k=2$. The enumeration of polymer coverings on two-dimensional rectangular lattices is considered as "intractable". We prove that the number of coverings of $s$ polymer satisfies a simple recurrence relation $\sum_{i=0}^{2s} (-1)^i \binom{2s}{i} a_{n-i, m-i} = 2^s {(2s)!} / {s!}$ on a $n \times m$ rectangular lattice with open boundary conditions in both directions.