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**The statistics of dimers on a lattice. I: The number of dimer arrangements on a quadratic lattice.**
*(English)*
Zbl 1244.82014

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.

### MSC:

82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |

### 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.