definition is essentially the same in the sense that there is a bijection from $\mathcal{D}_n(n+2k)$ to the set of standard Young tableaux of shape $\lambda=(n+k,k)$, which preserves the comajor statistics (see Section \[section33\]). However, the tranlation of the admissibility conditions seem to be quite technical and hence we prefer to work with the notion of Dyck paths instead of two row partitions.

The main results {#section4}
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In this section we summarize the main results of the paper. We have arranged it so that this section can be read essentially independently of the representation theory of $\mathfrak{sl}_2[t]$.

 {#section-5}

We will determine the polynomial in two steps. In the first step we will reduce the computation of graded multiplicities in Demazure flags of fusion products to the computation of graded multiplicities in local Weyl modules only. To be more precise, we will show that there exists a polynomial in finitely many variables such that a suitable evaluation of that polynomial at elements of the form $\mathcal{V}^{1\rightarrow \xi_i}_{\bullet,\bullet}(q)$ for $1\leq i\leq \ell$ gives the desired polynomial $\mathcal{V}^{\xi\rightarrow m}_{n}(q)$. In the second step we give a combinatorial formula for graded multiplicities in Demazure flags of local Weyl modules in terms of Dyck paths.

