Partitions with the same hook multiset

TL;DR

This paper investigates the conditions under which partitions sharing the same hook multiset are conjugate, providing a criterion to distinguish conjugate from non-conjugate partitions with identical hook multisets.

Contribution

It offers a necessary and sufficient criterion to determine when partitions with identical hook multisets are conjugate, addressing a question posed in 1977.

Findings

Provides a criterion for conjugacy based on hook multisets

Clarifies the relationship between hook multisets and partition conjugation

Answers a long-standing open question from 1977

Abstract

It is well-known that two conjugate partitions have the same hook multiset. But two different partitions with the same hook multiset may not be conjugate to each other. In $1977$, Herman and Chung proposed the following question: What are the necessary and sufficient conditions for partitions to be determined by their hook multisets up to conjugation? In this paper, we will answer this question by giving a criterion to determine whether two different partitions with the same hook multiset are conjugate to each other.