Identifying Young Diagrams Among Residue Multisets

Abstract

To any Young diagram we can associate the multiset of residues of all its nodes. This paper is concerned with the inverse problem: given a multiset of elements of \(\mathbb {Z}/e\mathbb {Z}\), does it comes from a Young diagram? We give a full solution in level one and a partial answer in higher levels for Young multidiagrams, using Fayers’s notions of core block and weight of a multipartition. We apply the result in level one to study a shift operation on partitions.