triangular unipotent whose entries are given by $f_{i,j}(\mathcal{V}),\  0\leq i,j\leq k,$ where $f_{i,j}(\mathcal{V})=\delta_{i,j}$ if $i\geq j$ and otherwise $$f_{i,j}(\mathcal{V})=\sum_{r\geq 0}(-1)^{r+1} \sum_{i=p_0<p_1<\cdots<p_{r}<p_{r+1}=j}\mathcal{V}_{p_0,p_1}\mathcal{V}_{p_1,p_2}\cdots \mathcal{V}_{p_r,p_{r+1}}.$$ Now the first entry of the vector $v=U^{-1}b$ gives the desired result.

Let $V(\xi)$ be a truncated Wey module, i.e. $\xi=((d+1)^j,d^{N-j})$ where $N,d,j\in {\mathbb Z}_+$ are such that $0<j<N$ and $|\xi|=dN+j$. Applying Proposition \[reductionpr\] to this setting gives $$\mathcal{V}_{|\xi|-2k}^{\xi\rightarrow \xi_0}(q)=\sum_{p=0}^k q^p\ \mathcal{V}_{(N-j)d,(N-j)d-2p}^{d \rightarrow d+1}(q)\ \mathcal{V}_{|\xi|-2p,|\xi|-2k}^{d+1 \rightarrow \xi_0}(q).$$

 {#section-6}

After the reduction in Proposition \[reductionpr\], our second result focussed on a combinatorial formula for the graded multiplicities in a level $m$–Demazure flag of a local Weyl module. Recall the definition of admissible Dyck paths from Definition \[maindef\].

\[mainthm1\] Let $m\in{\mathbb N}$, $n\in{\mathbb Z}_+$. We have, $$\mathcal{A}^{1\rightarrow m}_n(x,q)=\sum_{P\in \mathcal{D}_{m,n}}q^{\text{comaj}(P)}\ x^{d(P)},$$ where $d(P)$ denotes the number of down–steps of $P$. In particular, for any $s\in{\mathbb Z}_+$ we get $$\mathcal{V}^{1\rightarrow m}_{s,n}(q)=\sum_{P\in\mathcal{D}_{m,n}(s)}q^{\text{comaj}(P)}.$$

The proof of the above theorem will be postponed to Section \[section5\], but to avoid technical difficulties in the rest of the paper we will handle the case $m=1$ seperately. If $m=1$ we have $A(1,n)=\{(0,1),(1,2),\dots,(n-1,n)\}$ and hence $$\mathcal{D}_{1,n}=\{\underbrace{11\cdots 1}_{n}\}=\mathcal{D}_{1,n}(n).$$ This yields, $$\sum_{P\in \mathcal{D}_{1,n}(s)}q^{\text{comaj}(P)}=\delta_{s,n}$$ and the theorem follows immediately from . *So we will assume from now on that $m\geq 2.$*

If 