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Multiplicities of some maximal dominant weights of the $\widehat{s\ell}(n)$-modules $V(kΛ_0)$

For $n \geq 2$ consider the affine Lie algebra $\widehat{s\ell}(n)$ with simple roots $\{α_i \mid 0 \leq i \leq n-1\}$. Let $V(kΛ_0), \, k \in \mathbb{Z}_{\geq 1}$ denote the integrable highest weight $\widehat{s\ell}(n)$-module with highest weight $kΛ_0$. It is known that there are finitely many maximal dominant weights of $V(kΛ_0)$. Using the crystal base realization of $V(kΛ_0)$ and lattice path combinatorics we determine the multiplicities of a large set of maximal dominant weights of the form $kΛ_0 - λ^\ell_{a,b}$ where $ λ^\ell_{a,b} = \ellα_0 + (\ell-b)α_1 + (\ell-(b+1))α_2 + \cdots + α_{\ell-b} + α_{n-\ell+a} + 2α_{n - \ell+a+1} + \ldots + (\ell-a)α_{n-1}$, and $k \geq a+b$, $a,b \in \mathbb{Z}_{\geq 1}$, $\max\{a,b\} \leq \ell \leq \left \lfloor \frac{n+a+b}{2} \right \rfloor-1 $. We show that these weight multiplicities are given by the number of certain pattern avoiding permutations of $\{1, 2, 3, \ldots \ell\}$.