### Abstract

The method of Hock and McQuistan used recently to solve the occupation statistics for indistinguishable dumbbells (or dimers) on a 2×2×*N* lattice is extended further to obtain, for the *L*×*M*×*N* lattice, general expressions for the normalization, expectation, and dispersion of the statistics, and their limit as *N* becomes very large. In particular, an explicit expression of the partition function in the thermodynamic limit Ξ(*x*) is obtained for any value of the absolute activity *x* of dimers. The developed mathematical formalism is then applied to planar lattices, 1×*M*×*N*, with *M*=1, 2, 3, and 4. The known results for *M*=1 and 2 are recovered, and some new ones are obtained. The recurrence relation for the number *A*(*q*,*N*) of arrangements of *q* dumbbells on a 1×*M*×*N* lattice which has 3 and 5 terms when *M*=1 and 2, respectively, is found to have 15 and 65 terms for *M*=3 and 4. Analysis and extrapolation of the results enable one to predict the expectation 〈θ〉_{1M N } on a planar 1×*M*×*N* lattice to be 63.4%, in the limit as both *M* and *N* become infinite. We also find an upper bound on the quantity *M* *N*[〈θ^{2}〉−(〈θ〉)^{2}] in the limit as both *M* and *N* become infinite. In the thermodynamic limit (*M*&*N*→∞) the partition function Ξ(1), for the absolute activity *x*=1, is found to be equal to 1.95. By limiting the number *M* of rows of infinite extent (*N*→∞) to just 4, we find that the error in determining Ξ(1) for the infinite two‐dimensional lattice is just 4.5%. In this paper Ξ(*x*) is obtained for any value of the absolute activity *x* for *M*=1 and 2. A more thorough study of Ξ(*x*), and its fast convergence with increasing values of *M*, and applications will be presented in a forthcoming article.

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