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{\bf Fuliang Lu, Lianzhu Zhang and Fenggen Lin}
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{\bf Enumeration of Perfect Matchings of a Type of Quadratic Lattice on the Torus}
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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 $x_1,x_2,\cdots,x_m$
and $y_1,y_2,\cdots ,y_m$, respectively, where $x_i$ corresponds to
$y_i(i=1,2,\dots, m)$. We denote by $Q_{m,n,r}$, the graph obtained
from quadrilateral cylinder of length $m$ and breadth $n$ by adding
edges $x_iy_{i+r}$ ($r$ is a  integer, $0\leq r <m$ and $i+r$ is
modulo $m$). Kasteleyn had derived explicit expressions of the
number of perfect matchings for $Q_{m,n,0}$ [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 $Q_{m,n,r}$ by
enumerating Pfaffians.

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