Wide Partition Conjecture

The following definitions are from Wide Partitions, Latin Tableaux, and Rota's Basis Conjecture by Timothy Y. Chow et al.

Definition 1. An integer partition μ is a subpartition of λ if the multiset of parts of μ is a submultiset of the multiset of parts of λ

Definition 2. An integer partition λ is wide if μ ≥ μ' in dominance order for every subpartition μ of λ. (Here μ' denotes the conjugate of μ).

Definition 3. An integer partition λ is Latin if there exists a tableau T of shape λ such that for every i, the ith row of T contains {1,2, ... , &lambda_i}, and such that every column of T contains distinct integers.

Conjecture: λ is wide if and only if it is Latin. It is easy to show that if λ is Latin then λ is wide, but the converse remains open.