Honeycomb-lattice monomer-dimer mixtures

초록

Originally proposed to model the adsorption of diatomic molecules on crystal surfaces, the monomer-dimer model has since found extensive applications in statistical mechanics and combinatorics. In this study, we examine mixtures of monomers and dimers on planar honeycomb lattices, focusing on the derivation of a generating function that enumerates all possible configurations as a function of monomer activity. We highlight the relevance of the Hosoya index in this context and introduce an expression for the matching polynomial specific to honeycomb structures. Furthermore, we investigate lozenge tilings of semiregular hexagons, which correspond bijectively to dimer coverings of honeycomb graphs. In particular, we provide a detailed combinatorial analysis of symmetric lozenge tilings of a hexagon with side lengths (m, m, n, m, m, n).

키워드