How to determine a partition up to conjugation using multisets of hook lengths

If two partitions are conjugate, their multisets of hook lengths are the same. Then one may wonder whether the multiset of hook lengths of a partition determines a partition up to conjugation. The answer turns out to be no. However, we may add an extra condition under which a given multiset of hook lengths determines a partition uniquely up to conjugation. Herman-Chung, and later Morotti found such a condition. We give an alternative proof of Morotti's theorem and generalize it.