q^p,$$ where $[V(\xi):\tau_p^* D(m,n)]$ is the multiplicity of $\tau_p^* D(m,n)$ in a level $m$–Demazure flag of $V(\xi)$. In the first step, we reduce the computation of graded multiplicities in excellent filtrations of fusion products to the computation of graded multiplicities in local Weyl modules. To be more precise, we show that there exists a polynomial in finitely many variables such that a suitable evaluation of that polynomial at elements of the form $\mathcal{V}_{\bullet}(q)^{(1,1,\dots,1)\rightarrow \bullet}$ gives the desired polynomial $\mathcal{V}_n(q)^{\xi\rightarrow m}$ (see Poposition \[reductionpr\]). In the second step we give a combinatorial formula for $\mathcal{V}_{n}^{\xi\rightarrow m}(q)$ in terms of Dyck paths when $\xi$ is a hook partition. Let $m,m'\in{\mathbb N}$, $s\in{\mathbb Z}_+$ such that $m\geq m'$ and $\xi=(m',1^s)$ a hook partition. Then we have (see Theorem \[mainthm1\]) $$\mathcal{V}^{\xi\rightarrow m}_{n}(q)=\sum_{P} q^{\text{comaj(P)}},$$ where the sum runs over the set of admissible Dyck paths (see Definition\[maindef\]) from the origin $(0,0)$ to $(s+m',n)$ which start with $m'$ up–steps and never cross the line $y=\max\{m-1,n\}$.

The appearance of Dyck paths in this set up is quite unexpected and intriguing. A number of consequences flow from our new combinatorial formula. We briefly summarize the consequences below:

-   We find new combinatorial interpretations of Ramanujan’s fifth order mock 