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For $n \geq 2$ consider the affine Lie algebra $\widehat{s\ell}(n)$ withsimple roots $\{\alpha_i \mid 0 \leq i \leq n-1\}$. Let $V(k\Lambda_0), \, k\in \mathbb{Z}_{\geq 1}$ denote the integrable highest weight$\widehat{s\ell}(n)$-module with highest weight $k\Lambda_0$. It is known thatthere are finitely many maximal dominant weights of $V(k\Lambda_0)$. Using thecrystal base realization of $V(k\Lambda_0)$ and lattice path combinatorics wedetermine the multiplicities of a large set of maximal dominant weights of theform $k\Lambda_0 - \lambda^\ell_{a,b}$ where $ \lambda^\ell_{a,b} =\ell\alpha_0 + (\ell-b)\alpha_1 + (\ell-(b+1))\alpha_2 + \cdots +\alpha_{\ell-b} + \alpha_{n-\ell+a} + 2\alpha_{n - \ell+a+1} + \ldots +(\ell-a)\alpha_{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 certainpattern avoiding permutations of $\{1, 2, 3, \ldots \ell\}$.

Multiplicities of some maximal dominant weights of the $\widehat{s\ell}(n)$-modules $V(k\Lambda_0)$