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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.
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Nicolas Jacon (2026) studied this question.
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