Higher-order expansions for the entropy of a dimer or a monomer-dimer system on d-dimensional lattices

Recently, an expansion as a power series in 1/d has been presented for the specific entropy of a complete dimer covering of a d-dimensional hypercubic lattice. This paper extends from 3 to 10 the number of terms known in the series. Likewise, an expansion for the entropy, dependent on the dimer density p, of a monomer-dimer system, involving a sum ∑(k)a(k)(d)p(k), has been offered recently. We herein extend the number of known expansion coefficients from 6 to 20 for the hypercubic lattices of general dimensionality d and from 6 to 24 for the hypercubic lattices of dimensionalities d<5. We show that these extensions can lead to accurate numerical estimates of the p-dependent entropy for lattices with dimension d>2. The computations of this paper have led us to make the following marvelous conjecture: In the case of the hypercubic lattices, all the expansion coefficients a(k)(d) are positive. This paper results from a simple melding of two disparate research programs: one computing to high orders the Mayer series coefficients of a dimer gas and the other studying the development of entropy from these coefficients. An effort is made to make this paper self-contained by including a review of the earlier works.