Enumeration of Perfect Matchings of a Type of Quadratic Lattice on the Torus

A quadrilateral cylinder of length m and breadth n is the Cartesian product of a m-cycle(with m vertices) and a n-path(with n vertices). Write the vertices of the two cycles on the boundary of the quadrilateral cylinder as x1, x2, · · · , xm and y1, y2, · · · , ym, respectively, where xi corresponds to yi(i = 1, 2, . . . , m). We denote by Qm,n,r, the graph obtained from quadrilateral cylinder of length m and breadth n by adding edges xiyi+r (r is a integer, 0 6 r < m and i+r is modulo m). Kasteleyn had derived explicit expressions of the number of perfect matchings for Qm,n,0 [P.W. Kasteleyn, The statistics of dimers on a lattice I: The number of dimer arrangements on a quadratic lattice, Physica 27(1961), 1209–1225]. In this paper, we generalize the result of Kasteleyn, and obtain expressions of the number of perfect matchings for Qm,n,r by enumerating Pfaffians.