# Hard-hexagon configurations for fixed hexagon number

The grand-canonical partition function of the hard-hexagon model is given by $$Z_N = \sum_{n=0}^{N/3} z^n g(n,N) \,,$$ where $N$ is the number of sites, $n$ the number of hard hexagons, $z>0$ the activity and $g(n,N)$ is the number of ways of placing $n$ particles on the $N$ sites such that no two particles are together or adjacent.

The total number of configurations is simply the case $z=1$ and has been evaluated exactly by Baxter. Are there similar exact or approximate results for $g(n,N)$ itself (not its sum)?

Notice that, since $g(n,N)=0$ for $n>N/3$, $Z_N$ is actually the unilateral Z-transform of $g(n,N)$, in its geophysical definition. By obtaining $Z_N$ for various $z$ from Baxter's solution involving the Rogers-Ramanujan identities, one can perhaps -- at least numerically and approximately -- perform an inverse Z-transform to evaluate $g(n,N)$ for $N\rightarrow\infty$.