Size of Multipartitions through Level-Rank Duality

We give short purely combinatorial proofs for certain formulae relating the size of an $l$-partition with the one of its $e$-quotient in the sense of Uglov. We deduce a way to compute the size of the $e$-core of multipartitions and the $e$-weight of a multipartition. All these multipartitions naturally appear in the context of level-rank duality for Ariki-Koike algebras and Cherednik algebras.