Multiplicities of maximal weights of the $\widehat{s\mathrm{\ell }}(n)$-module $V(k{\mathrm{\Lambda }}_0)$

Consider the affine Lie algebra $\widehat{s\mathrm{\ell }}(n)$ with null root $\delta$, weight lattice $P$ and set of dominant weights $P^{+}$. Let $V(k{\mathrm{\Lambda }}_0),k\in {\mathbb{Z}}_{\ge 1}$ denote the integrable highest weight $\widehat{s\mathrm{\ell }}(n)$-module with level $k\ge 1$ highest weight $k{\mathrm{\Lambda }}_0$. Let $wt(V)$ denote the set of weights of $V(k{\mathrm{\Lambda }}_0)$. A weight $\mu \in wt(V)$ is a maximal weight if $\mu +\delta \in ̸wt(V)$. Let $ma{x}^{+}(k{\mathrm{\Lambda }}_0)=max(k{\mathrm{\Lambda }}_0)\cap P^{+}$ denote the set of maximal dominant weights which is known to be a finite set. In 2014, the authors gave the complete description of the set $ma{x}^{+}(k{\mathrm{\Lambda }}_0)$. In subsequent papers the multiplicities of certain subsets of $ma{x}^{+}(k{\mathrm{\Lambda }}_0)$ were given in terms of some pattern-avoiding permutations using the associated crystal base theory. In this paper the multiplicity of all the maximal dominant weights of the $\widehat{s\mathrm{\ell }}(n)$-module $V(k{\mathrm{\Lambda }}_0)$ are given generalizing the known results.