Dimer Coverings of 1-Slab Cubic Lattices

초록

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 x n x 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, ..., mn in S-mn rearranging each element at most to a neighboring position in the shape of the matrix whose entries are 1,2, ..., mn in usual order.

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