of importance to the Rogers–Ramanujan identities is the following. Consider $L+1$ points on a line labeled by $i=0,1,2,\ldots,L$. Assign to each point a height variable $\sigma_i$ which takes on the values $0$ or $1$. In addition the height variables satisfy the restrictions $\sigma_0=\sigma_L=0$ and $\sigma_i\sigma_{i+1}=0$. An allowed configuration of height variables for a given length $L$ is called a *path* of length $L$, and the set of all paths of length $L$ is denoted by ${\mathcal{D}}_L$. One can illustrate a path graphically by drawing all points $(i,\sigma_i)$ and connecting adjacent points by straight lines. An example for a path with $L=9$ is given in Figure \[fig path\].

(200,50)(0,0) (10,20)[(1,1)[20]{}]{} (50,20)[(-1,1)[20]{}]{} (90,20)[(1,1)[20]{}]{} (130,20)[(-1,1)[20]{}]{} (150,20)[(1,1)[20]{}]{} (190,20)[(-1,1)[20]{}]{} (8,9)[0]{} (28,9)[1]{} (48,9)[2]{} (68,9)[3]{} (88,9)[4]{} (108,9)[5]{} (128,9)[6]{} (148,9)[7]{} (168,9)[8]{} (188,9)[9]{} (10,20)[(1,0)[190]{}]{} (10,20) (30,40) (50,20) (70,20) (90,20) (110,40) (130,20) (150,20) (170,40) (190,20)

The condition $\sigma_i\sigma_{i+1}=0$ requires that the paths consist of a certain number of non-overlapping triangles (this condition comes directly from the condition in the two-dimensional hard hexagon model requiring that no two particles can be on adjacent sites). To each path $\sigma=(\sigma_0,\ldots,\sigma_L)$ one may assign an energy $E(\sigma)$ by summing up the positions of the peaks, that is $$E(\sigma)=\sum_{j=1}^L j \sigma_j.$$ The energy of the 