Identifying Young diagrams among residue multisets

TL;DR

This paper investigates the inverse problem of determining whether a multiset of residues corresponds to a Young diagram, providing complete solutions at level one and partial insights at higher levels, with applications to shift operations on partitions.

Contribution

It offers a comprehensive solution for identifying Young diagrams from residue multisets at level one and extends partial results to higher levels using core blocks and weights.

Findings

Complete characterization at level one.

Partial solutions for higher levels.

Application to shift operations on partitions.

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 Z/eZ, 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.