Beyond Rouquier partitions

We obtain closed formulas, in terms of Littlewood-Richardson coefficients, for the canonical basis elements of the Fock space representation of $U_v(\hat{\mathfrak{sl}}_e)$ which are labelled by partitions having 'locally small' $e$-quotients and arbitrary $e$-cores. We further show that, upon evaluation at $v=1$, this gives the corresponding decomposition numbers of the $q$-Schur algebras in characteristic $l$ (where $q$ is a primitive $e$-th root of unity if $l \ne e$ and $q=1$ otherwise) whenever $l$ is greater than the size of each constituent of the $e$-quotient.