Rank-unimodality of Young's lattice via explicit chain decomposition

Young’s lattice $L(m,n)$ consists of partitions having $m$ parts of size at most $n$, ordered by inclusion of the corresponding Ferrers diagrams. K. O’Hara gave the first constructive proof of the unimodality of the Gaussian polynomials by expressing the underlying ranked set of $L(m,n)$ as a disjoint union of products of centered rank-unimodal subsets. We construct a finer decomposition which is compatible with the partial order on Young’s lattice, at the cost of replacing the cartesian product with a more general poset extension. As a corollary, we obtain an explicit chain decomposition which exhibits the rank-unimodality of $L(m,n)$. Moreover, this set of chains is closed under the natural rank-flipping involution given by taking complements of Ferrers diagrams.