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

TL;DR

This paper provides an explicit chain decomposition of Young's lattice that demonstrates its rank-unimodality, extending previous constructive proofs and revealing symmetry properties under Ferrers diagram complements.

Contribution

It introduces a finer, order-compatible decomposition of Young's lattice, replacing Cartesian products with a more general poset extension, to explicitly prove rank-unimodality.

Findings

Constructed a chain decomposition demonstrating rank-unimodality.

Established the set of chains is closed under rank-flipping involution.

Extended previous proofs by providing an explicit, order-compatible decomposition.

Abstract

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.