Chain Decompositions of q, t-Catalan Numbers: Tail Extensions and Flagpole Partitions
Chain Decompositions of q, t-Catalan Numbers: Tail Extensions and Flagpole Partitions
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q, t-加泰罗尼亚数的链分解：尾部扩展和旗杆分区
DOI：
10.1007/s00026-022-00590-7
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发表时间：
2022
影响因子：
0.5
通讯作者：
Loehr, Nicholas A.
中科院分区：
文献类型：
--
作者：
Han, Seongjune;Lee, Kyungyong;Li, Li;Loehr, Nicholas A.
这篇文章是正在进行的q，t-卡塔兰数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deficitk的无限分割集分解为由sizek的分割索引的链的不相交并集。在其他结构特性中，这些链可以配对以改进著名的对称性。之前，我们介绍了一个构建每条链尾部的map。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建设chainfor（广义）旗杆分区，假设链索引的某些特定的较小的分区（取决于）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。
This article is part of an ongoing investigation of the combinatorics ofq,t-Catalan numbers. 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 deficitkinto a disjoint union of chainsindexed by partitions of sizek. Among other structural properties, these chains can be paired to give refinements of the famous symmetry property. Previously, we introduced a map that builds the tail part of each chain. 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 chainfor a (generalized) flagpole partition, assuming that the chains indexed by certain specific smaller partitions (depending on) are already known. We also give some enumerative and asymptotic results for flagpole partitions and their generalized versions.