Kasteleyn, P. W. The statistics of dimers on a lattice. I: The number of dimer arrangements on a quadratic lattice. (English) Zbl 1244.82014 Physica 27, 1209-1225 (1961). Summary: The number of ways in which a finite quadratic lattice (with edges or with periodic boundary conditions) can be fully covered with given numbers of “horizontal” and “vertical” dimers is rigorously calculated by a combinatorial method involving Pfaffians. For lattices infinite in one or two dimensions asymptotic expressions for this number of dimer configurations are derived, and as an application the entropy of a mixture of dimers of two different lengths on an infinite rectangular lattice is calculated. The relation of this combinatorial problem to the Ising problem is briefly discussed. Cited in 14 ReviewsCited in 422 Documents MSC: 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics Keywords:dimer arrangements; quadratic lattice × Cite Format Result Cite Review PDF Full Text: DOI Online Encyclopedia of Integer Sequences: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. Octagonal pyramidal numbers: a(n) = n*(n+1)*(2*n-1)/2. Number of domino tilings (or dimer coverings) of a 2n X 2n square. Number of ways of covering a 2n X 2n lattice by 2n^2 dominoes with exactly 4 horizontal (or vertical) dominoes. Number of ways of covering a 2n X 2n lattice with 2n^2 dominoes of which exactly 6 are horizontal (or vertical) dominoes. Decimal expansion of growth constant C for dimer model on square grid. Array T(m,n) read by antidiagonals: number of domino tilings (or dimer tilings) of the m X n grid (or m X n rectangle), for m>=1, n>=1. Number of perfect matchings in the graph C_7 X C_{2n}. Number of perfect matchings in the graph C_5 X C_{2n}. Number of perfect matchings in the graph C_6 x C_n. Number of domino tilings of a 16 X n rectangle. Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards: number of perfect matchings in the graph C_{2n} x C_k. Number of domino tilings of a 32 X n rectangle. Irregular triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n^2) is the number of ways to tile an n X n chessboard with k rook-connected polyominoes of equal area. Irregular triangle read by rows: T(n,k) (n >= 1, 1 <= k <= number of divisors of n^2) is the number of ways to tile an n X n chessboard with d_k rook-connected polyominoes of equal area, where d_k is the k-th divisor of n^2.