Sách Enumeration of perfect matchings of a type of quadratic lattice on the torus

Abstract
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.
Keywords: Pfaffian; Perfect matching; Quadratic lattice; Torus.