Two counterparts of the TFK formula for cylinder graphs

Abstract

The classical 1961 solution to the problem of determining the number of perfect matchings (or dimer coverings) of a rectangular grid graph — due independently to Temperley and Fisher, and Kasteleyn, who found the TFK formula — consists of changing the sign of some of the entries in the adjacency matrix so that the Pfaffian of the new matrix gives the number of perfect matchings, and then evaluating this Pfaffian. Another classical method is to use the Lindström-Gessel-Viennot theorem on non-intersecting lattice paths to express the number of perfect matchings as a determinant, and then evaluate this determinant. In this paper we present a new method, which relies on the Cauchy-Binet theorem, and use it to solve the two dimensional dimer problem for cylinder graphs on the square and on the hexagonal lattice.We provide explicit product formulas for these two kinds of cylinder graphs. One advantage of our formula for the square lattice compared to the TFK formula is that ours has a linear number of factors, while the number of factors in the former is quadratic. Our result for the hexagonal lattice yields a formula for the number of periodic stepped surfaces that fit in an infinite tube of given cross-section. This can be regarded as a counterpart of MacMahon's boxed plane partition theorem.