Lattice gas generalization of the hard hexagon model. I. Star-triangle relation and local densities

In the solvable hard hexagon model there is at most one particle in every pair of adjacent sites, and the solution automatically leads to various mathematical identities, in particular to the Rogers-Ramanujan relations. These relations have been generalized by Gordon. Here we construct a solvable model with at most two particles per pair of adjacent sites, and find the solution involves the next of Gordon's relations. We conjecture the corresponding solution for a model with at most n particles per pair of adjacent sites: this involves all Gordon's relations, as well as others that we will discuss in a subsequent paper.