q，t-Catalan数的链分解：尾扩张和旗杆分拆
Chain decompositions of q,t-Catalan numbers: tail extensions and flagpole partitions
摘要：
这篇文章是正在进行的$q,t$ -Catalan数$\textrm{Cat}_n(q,t)$的组合学研究的一部分。我们开发了一个结构理论的基础上的分区统计dinv，赤字，和最小三角形高度的整数分区。我们的目标是将赤字$k$的无限分区集分解为由大小为$k$的分区索引的链$\mathcal{C}_μ$的不相交并集。在其他结构性质中，这些链可以配对以给出著名的对称性质$\textrm{Cat}_n(q,t)=\textrm{Cat}_n(t,q)$的改进。在前面，我们介绍了构建每条链的尾部$\mathcal{C}_μ$的映射。我们在这里的第一个主要贡献是扩展这个映射，为每条链构建更大的二阶尾。其次，我们引入了新的划分类，称为旗杆划分和广义旗杆划分。第三，我们描述了一个递归的构造链$\mathcal{C}_μ$（广义）旗杆分区$μ$，假设链索引的某些特定的较小的分区（取决于$μ$）是已知的。我们也给出了旗杆分拆及其推广形式的一些计数和渐近结果。
英文摘要：
This article is part of an ongoing investigation of the combinatorics of $q,t$-Catalan numbers $\textrm{Cat}_n(q,t)$. We develop a structure theory for integer partitions based on the partition statistics dinv, deficit, and minimum triangle height. Our goal is to decompose the infinite set of partitions of deficit $k$ into a disjoint union of chains $\mathcal{C}_μ$ indexed by partitions of size $k$. Among other structural properties, these chains can be paired to give refinements of the famous symmetry property $\textrm{Cat}_n(q,t)=\textrm{Cat}_n(t,q)$. Previously, we introduced a map that builds the tail part of each chain $\mathcal{C}_μ$. Our first main contribution here is to extend this map to construct larger second-order tails for each chain. Second, we introduce new classes of partitions called flagpole partitions and generalized flagpole partitions. Third, we describe a recursive construction for building the chain $\mathcal{C}_μ$ for a (generalized) flagpole partition $μ$, assuming that the chains indexed by certain specific smaller partitions (depending on $μ$) are already known. We also give some enumerative and asymptotic results for flagpole partitions and their generalized versions.
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