path in Figure \[fig path\] is $E(\sigma)=1+5+8=14$. The generating function of paths of length $L$ which is also called a one dimensional configuration sum is defined as $$\label{path}
X(L)=\sum_{\sigma\in {\mathcal{D}}_L} q^{E(\sigma)}.$$ The path picture immediately implies that $X(L)$ satisfies the following initial conditions and recurrence which completely specify it $$\label{rec}
\begin{split}
X(0)&=X(1)=1\\
X(L)&=X(L-1)+q^{L-1}X(L-2).
\end{split}$$

The aim is to find explicit expressions for $X(L)$. We will describe two ways to obtain such an expression which will be related to the two sides of .

Fermionic formula {#subsec:fermi}
-----------------

Interpret each peak in a path as a particle, that is, there is a particle at site $i$ if $\sigma_i=1$. Fix the number of particles to be $n$. The ground state path ${\sigma_{\mathrm{G}}}$ with minimal energy is the path with particles at positions $1,3,\ldots,2n-1$. The energy of ${\sigma_{\mathrm{G}}}$ is $E({\sigma_{\mathrm{G}}})=1+3+\cdots+2n-1=n^2$. An example of the ground state path with $3$ particles is shown in Figure \[fig:ground\].

(200,50)(0,0) (10,20)[(1,1)[20]{}]{} (50,20)[(-1,1)[20]{}]{} (50,20)[(1,1)[20]{}]{} (90,20)[(-1,1)[20]{}]{} (90,20)[(1,1)[20]{}]{} (130,20)[(-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) (70,40) (90,20) (110,40) (130,20) (150,20) (170,20) (190,20)

All other paths with $n$ particles can be obtained from ${\sigma_{\mathrm{G}}}$ by moving particles to the right in such a 