$B \otimes B_{\ell{\Lambda}_0}$. The affine highest weight vectors of weight ${\Lambda}'$, whose number is the above multiplicity, are the summands of ${\overline{X}}^\ell(B,{\Lambda})$.

It will be shown in sections \[sec:bose\] and \[sec:fermi\] that there are two different evaluations of ${\overline{X}}(B,{\Lambda})$ and ${\overline{X}}^\ell(B,{\Lambda})$ which give rise to $q$-identities.

Bosonic evaluation {#sec:bose}
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Here we present the bosonic evaluation of ${\overline{X}}(B,\Lambda)$ and ${\overline{X}}^\ell(B,\Lambda)$ as defined in and . Similarly to the bosonic evaluation of $X(L)$ of section \[subsec:bose\], the bosonic evaluation of ${\overline{X}}(B,\Lambda)$ and ${\overline{X}}^\ell(B,\Lambda)$ can be obtained by inclusion-exclusion arguments as shown in [@SS]. We discuss the main ideas and techniques of sign-reversing involutions.

Classically-restricted case
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Let $B=B_L\otimes \cdots \otimes B_1$ be a $U_q({\mathfrak{g}})$-crystal and let ${\Lambda}$ be a classical weight. The Weyl group $W$ of ${\mathfrak{g}}$ which is generated by the simple reflections $r_i$ as in with $i$ restricted to $i\in J$. The bosonic expression for the generating function of classically-restricted paths ${\overline{X}}(B,{\Lambda})$ as defined in is given by $$\label{bose cr}
{\overline{X}}(B,{\Lambda})=\sum_{\omega \in W} (-1)^\omega \; 
 {\overline{S}}(B,\omega({\Lambda}+\rho)-\rho)$$ where $(-1)^\omega$ is the sign of $\omega$ and $\rho$ is half the sum of all positive roots. This formula follows directly from Weyl’s character formula.

As a warm-up for the level-restricted case, we 