Dimer Coverings of 1-Slab Cubic Lattices

Seungsang Oh

Abstract: Lattice dimer statistics in two-dimension gained momentum in 1961 when the exact solution of the enumeration of pure dimer coverings of a rectangular square lattice was found by Kasteleyn (Physica 27:1209–1225) and Temperley and Fisher (Philos Magn 6:1061–1063). Recently the author introduced the state matrix recursion method to generate the partition function for monomer–dimer coverings in the square lattice with monomer and dimer activities. In this paper, stepping up a dimension, we apply this method to the enumeration of pure dimer coverings in the 1-slab cubic \(m \times n \times 2\) lattice. Its asymptotic behavior is also derived. As a pure dimer covering in two-dimension is known as a perfect matching or a domino tiling, a pure dimer covering in the 1-slab cubic lattice is considered as a perfect matching or a packing of a three-dimensional 2-layer box with dicubes which are formed by two unit cubes joined face to face. We further discuss a new bijective relation of this model to the set of so-called matrix entry neighbor-permutations of \(1, 2, \dots , mn\) in \(S_{mn}\) rearranging each element at most to a neighboring position in the shape of the matrix whose entries are \(1, 2, \dots , mn\) in usual order.