Regular structures of an intractable enumeration problem: a diagonal recurrence relation of monomer-polymer coverings on two-dimensional rectangular lattices
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909054403608576 |
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| author | Kong, Yong |
| author_facet | Kong, Yong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04147 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regular structures of an intractable enumeration problem: a diagonal recurrence relation of monomer-polymer coverings on two-dimensional rectangular lattices Kong, Yong Combinatorics Statistical Mechanics 05A15 (Primary) 82B20, 03D15 (Secondary) F.1.3 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. |
| title | Regular structures of an intractable enumeration problem: a diagonal recurrence relation of monomer-polymer coverings on two-dimensional rectangular lattices |
| topic | Combinatorics Statistical Mechanics 05A15 (Primary) 82B20, 03D15 (Secondary) F.1.3 |
| url | https://arxiv.org/abs/2411.04147 |