and their generating functions are discussed in section \[sec:crystal\]. The two different ways to evaluate the paths generating functions are subject of sections \[sec:bose\] and \[sec:fermi\], respectively. In particular in section \[sec:lev C\] we present new fermionic formulas for level-restricted paths. We close in section \[sec:open\] with some outstanding open problems.

Before indulging in the fascinating theory of crystal bases there is one important point that needs to be addressed. Applying Jacobi’s triple product identity $$\sum_{n=-\infty}^\infty z^n q^{n^2} =
 \prod_{n=0}^\infty (1-q^{2n+2})(1+z q^{2n+1})(1+z^{-1} q^{2n+1})$$ the right-hand sides of and can be rewritten as alternating sums yielding the identities $$\begin{aligned}
\label{bf1}
\sum_{n=0}^\infty \frac{q^{n^2}}{(q)_n} &=
 \frac{1}{(q)_\infty} \sum_{j=-\infty}^\infty (-1)^j q^{\frac{j}{2}(5j+1)}\\
\label{bf2}
\sum_{n=0}^\infty \frac{q^{n(n+1)}}{(q)_n} &=
 \frac{1}{(q)_\infty} \sum_{j=-\infty}^\infty (-1)^j q^{\frac{j}{2}(5j+3)}.\end{aligned}$$ The two different evaluations of generating functions of crystal paths that were mentioned above really yield (polynomial) analogues of and rather than and . Identities relating sums to alternating sums are more general than identities relating sums to products. Only in special cases can the alternating sums be evaluated as products, namely when Jacobi’s triple product identity or more generally the Macdonald identities [@Mac] can be applied.

Acknowledgements {#acknowledgements .unnumbered}
----------------

We are grateful to Tim Baker for discussions. A.S. would like to thank the organizers 