Bruce Berndt and Ken Ono for the invitation to present these results at the conference on $q$-Series with Applications to Combinatorics, Number Theory, and Physics held in Urbana-Champaign in October 2000. Thanks for this exciting conference!

The Rogers–Ramanujan identities and the Hard Hexagon model {#sec:HHM}
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The hard hexagon model is a two-dimensional lattice model of a gas of hard or non-overlapping particles. The particles are placed on a triangular lattice such that no two particles can occupy two adjacent sites. If one views each particle as the center of a hexagon, then the condition that no two particles can be adjacent translates into the condition that the hexagons cannot overlap. This explains the name of the model. An example of an allowed particle configuration is shown in Figure \[fig:hhconfig\].

(260,140)(0,0) (0,0)(260,0) (0,34)(260,34) (0,67)(260,67) (0,102)(260,102) (0,134)(260,134) (0,67)(40,134) (0,0)(80,134) (40,0)(120,134) (80,0)(160,134) (120,0)(200,134) (160,0)(240,134) (200,0)(260,102) (240,0)(260,34) (40,0)(0,67) (80,0)(0,134) (120,0)(40,134) (160,0)(80,134) (200,0)(120,134) (240,0)(160,134) (260,34)(200,134) (260,102)(240,134) (60,102)[4]{}[Black]{}[Black]{} (60,34)[4]{}[Black]{}[Black]{} (120,67)[4]{}[Black]{}[Black]{} (200,67)[4]{}[Black]{}[Black]{} (40,67)(80,67) (80,67)(100,102) (100,102)(80,134) (80,134)(40,134) (40,134)(20,102) (20,102)(40,67) (40,0)(80,0) (80,0)(100,34) (100,34)(80,65) (80,65)(40,65) (40,65)(20,34) (20,34)(40,0) (102,34)(140,34) (140,34)(160,67) (160,67)(140,102) (140,102)(102,102) (102,102)(82,67) (82,67)(102,34) (180,34)(220,34) (220,34)(240,67) (240,67)(220,102) (220,102)(180,102) (180,102)(160,67) (160,67)(180,34)

To pack the lattice densely all particles must lie on one of the three sublattices corresponding to the three corners 