$n=3$, $m=2$ and $s=5$ we have $A(2,3)=\{(0,2),(2,3)\}$ and hence $\mathcal{D}_{2,3}(5)=\{10111, 11011\}$. We get $\mathcal{V}_{5,3}^{1\rightarrow 2}(q)=q^3+q^4$.

(0,0) grid (5,3); (0,0) – (1,1)–(2,0) –(5,3);

(0,0) grid (5,3); (0,0) – (2,2) – (3,1)–(5,3);

\[fig3\]

We shall prove in Section \[section5\] a slightly more general version of Theorem \[mainthm1\], namely we will give a combinatorial formula for the polynomials in terms of Dyck paths for every hook partition $\xi$ (see Theorem \[hook\]).

 {#section-7}

Before we proceed to the proof of Theorem \[mainthm1\] we will discuss several consequences. For simple finite-dimensional Lie algebras of non–simply laced type local Weyl modules are in general not isomorphic to level one Demazure modules. Nevertheless, Naoi proved in [@Na11 Sections 4 and 9] that a local Weyl module (see for example [@Na11 Definition 3.1] for a precise definition) admits a level one Demazure flag. If ${\mathfrak{g}}$ is of type $B_n$ or $G_2$ one can use the $\mathfrak{sl_2}$–theory (in particular, the combinatorial formula in Theorem \[mainthm1\]) to give combinatorial formulas for $\sum_p [W(\lambda): \tau_p^* D(1,\mu)]q^p$, where $\lambda,\mu$ are dominant integral ${\mathfrak{g}}$–weights. This fact follows from an inspection of the proof given in [@Na11] and a precise statement can be found in [@CSSW14 Proposition 2.5].

Let ${\mathfrak{g}}$ of type $B_n$ and 