way that the particles never overtake each other. If the length of the path is $L$, the rightmost particle can move at most $L-2n$ positions to the right. Hence the paths of length $L$ with $n$ particles are in one-to-one correspondence with partitions with at most $n$ parts not exceeding $L-2n$. If a path $\sigma$ corresponds to partition ${\lambda}$ then its energy is $E(\sigma)=E({\sigma_{\mathrm{G}}})+|{\lambda}|=n^2+|{\lambda}|$. The generating function of partitions with at most $n$ parts not exceeding $m$ is the $q$-binomial coefficient defined as $${\genfrac{[}{]}{0pt}{}{m+n}{n}}=\frac{(q)_{m+n}}{(q)_m(q)_n}$$ for $m,n\in{\mathbb{N}}$ and zero otherwise. This implies the following explicit expression for $X(L)$ $$\label{RR fermi}
X(L)=\sum_{n=0}^\infty q^{n^2} {\genfrac{[}{]}{0pt}{}{L-n}{n}}.$$ Because of its interpretation in terms of non-overlapping particles this expression is called a quasiparticle or fermionic formula.

Bosonic formula {#subsec:bose}
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As opposed to there is another expression for $X(L)$ given by $$\label{RR bose}
X(L)=\sum_{j=-\infty}^\infty (-1)^j q^{\frac{j}{2}(5j+1)}
 {\genfrac{[}{]}{0pt}{}{L}{\lfloor \frac{L-5j}{2}\rfloor}}$$ where $\lfloor x\rfloor$ is the largest integer not exceeding $x$. It can be proven by showing that the right-hand side satisfies the recurrence and initial condition .

However, formula can also be interpreted in terms of paths. For this purpose the above path picture needs to be slightly reformulated. The state diagram of the paths in the hard 