$m=2$ or $G_2$ and $m=3$. Further denote by $\alpha$ the unique simple short root with coroot $h_{\alpha}$. Let $\lambda,\mu$ two dominant integral weights of ${\mathfrak{g}}$. We have $$[W(\lambda): \tau_p^* D(1,\mu)]= \begin{cases}|\{P\in \mathcal{D}_{m,\mu(h_{\alpha})}(\lambda(h_{\alpha})): \text{comaj}(P)=p\}|,& \lambda-\mu\in {\mathbb Z}_+\alpha\\
0,& \text{otherwise.}
\end{cases}$$

 {#section-8}

In the special case when $m=2$, we can derive another combinatorial description of the graded multiplicities in terms of bounded partitions which we record in the next lemma. This result can also be obtained by a straightforward calculation combining [@CSSW14 Theorem 3.3] and [@And98 Chapter 3] as pointed out in [@MJ16 Section 4.3]. Our proof is different and uses only the combinatorial formula stated in Theorem \[mainthm1\]. For integers $a,b,c\in{\mathbb Z}$, denote by $\rho^{a}_{b}(c)$ the set of partitions of $c$ with at most $a$ parts such that each part is bounded by $b$. Recall the integers $n_0,n_1$ from Section \[section23\]

\[lpart\] Let $k,n\in {\mathbb Z}_+$. Then we have $$\mathcal{V}^{1\rightarrow 2}_{n+2k,n}(q)=\sum^{k(k+n)}_{\ell=0}   |\rho^{k}_{n_1}(k(k+n)-\ell)|\hspace{0,04cm} q^{\ell}.$$

If $0\leq n\leq 1$ we have $n_1=0$ and hence the right hand side is equal to $q^{k^2}$. Since each path in $\mathcal{D}_{n}^{1}(n+2k)$ has height $\leq 1$ the same is true for the left hand side and the lemma is immediate; so assume that $n>1$. Let $P\in \mathcal{D}_{2,n}(n+2k)$ such 