Classical freeness of ${\widehat{\mathfrak{s}\mathfrak{l}}}_n$ at level $1$ via combinatorics

We use a family of Rogers–Ramanujan-type combinatorial identities of Dousse–Konan involving coloured partitions to prove classical freeness of the simple vertex operator algebras based on ${\widehat{\mathfrak{s}\mathfrak{l}}}_n$ at level $1$. These identities are used to produce Gröbner bases for the relevant arc algebras.