2021/04/22 by Guoce Xin, Yueming Zhong, Xin, Guoce +1
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.2104.11003
Let L(m,n) denote Young's lattice, consisting of all partitions whose Young diagrams are contained within an m× n rectangle. It is a classical result that the partially ordered set L(m,n) is rank-symmetric, rank-unimodal, and Sperner; however, finding a direct combinatorial proof via an explicit order matching remains a prominent open problem in the field. In this paper, we address this challenge by constructing explicit order matchings for L(3,n) and extending our methods to comprehensively cover L(4,n). To achieve this, we introduce a novel ``chain tableau" representation, which serves as a powerful tool for identifying and characterizing complex combinatorial patterns. Notably, we demonstrate that the same order matchings can be independently derived using both a greedy algorithm and a recursive kneading process. This work not only resolves the explicit matching problem for m=3 and m=4 but also establishes robust structural tools that may offer valuable insights into the general L(m,n) case.