Unimodal and Sperner ideals in Young's lattice

Given a Young diagram D, we consider the lattice Y(D) consisting of Young diagrams that fit inside D. A famous theorem of Sylvester implies that Y(D) is unimodal if D is a rectangle. O'Hara gave the first combinatorial proof of this fact via an ingenious recursive argument. However, Stanton showed that Y(D) is not unimodal in general. We discuss a partial extension of O'Hara's construction to Y(D) for arbitrary D and its implications for unimodality and the Sperner property. We also discuss how one might adapt this approach to higher-dimensional partitions.