*Abstract:* The number of configurations of dimers on a large periodic lattice cube that cover a fraction p of the vertices is asymptotically of the form of the exponential of a function of p times the volume V, as V goes to infinity. This function is commonly called the entropy. For hyper-cubical lattices in d dimensions we establish an expression for this function of p, for small p, involving a (convergent) power series in p. With the conjecture that all the terms in this series are positive, confirmed by computation for the first 20 terms in all dimensions, the series is valid for 0
By either a spark of creative incite, or a fortuitous fumbling about, one was lead to consider for the monomer-dimer problem on connected regular bipartite graphs the quantity d(i)=ln(N(i)/r^i)-ln(N'(i)/(v-1)^i) where v is the number of vertices, r is the degree, and N(i) and N'(i) are the number of i-dimer configurations on the graph and on the complete graph with the same set of vertices, respectively. This quantity for a sequence of graphs converging to the lattice converges to a multiple of the sum in the entropy expression, if we set p ~ (2i/v). Then the fact that all derivatives of the sum in the entropy are positive (the conjecture holding) leads one to conjecture that all finite difference derivatives of d(i) are positive. The Virial series conjecture leads to a putative upper bound for the same finite difference derivatives. In fact it seems to be true that "almost all" graphs satisfy these lower and upper finite difference bounds! Looking just at the 0th derivative case one gets new upper and lower conjectured bounds on N(i), higher derivatives yielding a rich set of new bounds to study.