Chain Decompositions of q, t-Catalan Numbers via Local Chains
Chain Decompositions of q, t-Catalan Numbers via Local Chains
复制标题
通过本地链对 q, t-Catalan 数进行链分解
DOI：
10.1007/s00026-020-00512-5
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发表时间：
2020
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.
q，t-Catalan数通过其dinv和外部面积统计来枚举包含在三角形中的整数分区。Lee等人的论文（SIAM J Discr Math 32：191-232，2018）提出了一种理解对称性的新方法，该方法基于将所有整数划分的集合分解为无限链。每一个这样的全局链都有一个相反的链;这些联合收割机结合起来，给出了一个新的在qandt上对称的小切片。在这里，我们通过开发一种新的通用方法来推进Lee等人（SIAM J Discr Math 32：191-232，2018）的议程，该方法用于从称为局部链的较小元素构建全局链。我们定义了一个局部链的局部相反性质，它隐含了全局链所需要的相反性质。与相应的全局属性相比，此局部属性在特定情况下更容易验证。我们应用这一机制来构建所有的全球链的分区赤字最多。这证明了对于alln，次数至少为$$\left（{\开始{array}{c}n\\ 2\end{array}}\right）-11 $$的项在qandt中是对称的。
Theq,t-Catalan numberenumerates integer partitions contained in antriangle by their dinv and external area statistics. The paper by Lee et al. (SIAM J Discr Math 32:191–232, 2018) proposed a new approach to understanding the symmetry propertybased on decomposing the set of all integer partitions into infinite chains. Each such global chainhas an opposite chain; these combine to give a new small slice ofthat is symmetric inqandt. Here, we advance the agenda of Lee et al. (SIAM J Discr Math 32:191–232, 2018) by developing a new general method for building the global chainsfrom smaller elements calledlocal chains. We define alocal opposite propertyfor local chains that implies the needed opposite property of the global chains. This local property is much easier to verify in specific cases compared to the corresponding global property. We apply this machinery to construct all global chains for partitions with deficit at most. This proves that for alln, the terms inof degree at least $$\left( {\begin{array}{c}n\\ 2\end{array}}\right) -11$$ are symmetric inqandt.