对相邻部分之商限制的分拆和有序分拆的若干问题研究

Integer partition is one of the most important research subjects in the area of enumerative combinatorics. For centuries, it has attracted the attention of many famous mathematicians and developed rapidly since it has so many applications in the areas of q-series, number theory, group theory and physics..This project aim to study partitions and compositions constrained by the ratio of consecutive parts. We will try to obtain some new enumerative results concerned with this type of partitions and compositions by constructing bijections and involutions or by using MacMahon's partition analysis, and find some refinement of these results by considering some restrictions on the largest parts or other statistics. On the other hand, we will consider the connection between partitions and compositions constrained by the ratio of consecutive parts and the combinatorial representation of the Coxter group, permutations, infinite permutations, and the distributions of some permutation statistics. We will also calculate the generating functions of these partitions and compositions by considering some q-series identities, such as Rogers-Ramanujan identity, q-Gauss summation. Some geometrical problems such as the volumn of polytopes, lattice point enumeration will also be studied.

整数分拆作为组合数学中的一个基本而重要的研究对象，因其与q-级数，数论，群论，物理学等领域的紧密联系以及在其中的广泛应用，几个世纪以来，受到众多数学家的关注和重视并不断发展。.本项目着重围绕对相邻部分之商限制的分拆和有序分拆进行研究，将通过构建双射对合证明，利用MacMahon分拆分析方法等，试图得到新的与之相关的计数结果，以及已有结果的细化和推广形式。项目还将关注对相邻部分之商限制的分拆和有序分拆与Coxter群的组合表示，排列，无限排列等之间的关系，研究与之相关的各种排列统计量的分布。我们还将把对此类分拆生成函数的研究与Rogers-Ramanujan型等式，q-Gauss和式等q-级数等式结合起来。另外，项目将研究与此类分拆相关的格点计数，多面体体积等几何问题。