2013/03/11 by Vivek Dhand, Dhand, Vivek
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #math.CO
paper · pdf · doi:10.48550/arxiv.1303.2536
18 pages
arxiv created 2013/03/11 · openalex publication_date 2013/03/11 · arxiv updated 2013/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.