发表机构
Tianjin University（天津大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文推广Andrews-Dastidar划分配对理论至$k$-配对，通过两个保权双射给出配对指标-宽度分布与负秩枚举的组合解释，并扩展至Young图元组等价类与平面划分。
AI 中文摘要
我们将Andrews和Dastidar的划分配对理论加以推广，用$k$个相等部分的组来替代配对。两个保持权重的双射给出了联合配对指标-宽度分布和负秩枚举的组合解释。第一个双射将共轭与Stockhofe-Keith对应相结合，分别将$k$-配对指标和宽度映射到部分个数和最大部分。因此，它们的联合分布与$k$无关，并由一个高斯多项式给出。我们还获得了记录残差统计的有限细化。第二个双射将负$k$-配对秩的简单$k$-配对划分映射到带有标记内部间隙的$k$-正则划分。在固定的非零剩余类中，符号相消留下矩形划分，而间隙标记对应于最小部分未被划线的上划线选择。这给出了除数计数和上划线划分枚举中因子$1/2$的直接组合解释，推广了Andrews和Dastidar的奇除数和奇上划线结果。最后，受Andrews和Dastidar的对角配对的启发，我们将构造推广到$k$个Young图的有序元组。将双翼转移操作应用于分量图对，在这些元组上定义了一个等价关系。我们证明两个元组等价当且仅当它们具有相同的单元重数函数，并且每个等价类包含一个唯一的分量图嵌套的代表。我们还确定了等价类的基数，并将嵌套代表与矩形形状的平面划分等同起来。
英文摘要
We extend the partition pairing theory of Andrews and Dastidar by replacing pairs with groups of $k$ equal parts. Two weight-preserving bijections give combinatorial interpretations of the joint pairing index--width distribution and the negative-rank enumerations. The first combines conjugation with the Stockhofe--Keith correspondence and sends the $k$-pairing index and width to the number of parts and the largest part, respectively. Consequently, their joint distribution is independent of $k$ and is given by a Gaussian polynomial. We also obtain finite refinements that record the residual statistics. The second bijection sends simply $k$-paired partitions of negative $k$-pairing rank to $k$-regular partitions with marked internal gaps. In a fixed nonzero residue class, sign cancellation leaves rectangular partitions, while gap markings correspond to overlining choices with the smallest part not overlined. This gives direct combinatorial explanations of the divisor counts and the factor $1/2$ in the overpartition enumeration, extending the odd-divisor and odd-overpartition results of Andrews and Dastidar. Finally, motivated by the diagonal pairing of Andrews and Dastidar, we extend the construction to ordered tuples of $k$ Young diagrams. Applying the two-wing transfer operation to pairs of component diagrams defines an equivalence relation on these tuples. We show that two tuples are equivalent if and only if they have the same cell-multiplicity function, and that every equivalence class contains a unique representative whose component diagrams are nested. We also determine the cardinalities of the equivalence classes and identify the nested representatives with plane partitions of rectangular shape.
Comments23 pages
