B型和D型环形非交叉分拆的偏序集
Posets of annular non-crossing partitions of types B and D
摘要：
研究了B型环形非交叉置换集$\sncb (p,q)$，并引入了相应的B型环形非交叉分拆集$\ncb (p,q)$，其中$p$和$q$是两个正整数.证明了$\sncb (p,q)$和$\ncb (p,q)$之间的自然双射是偏序集同构，其中$\sncb (p,q)$上的偏序是由超八面体群$B_{p+q}$导出的，而$\ncb (p,q)$上的偏序是由逆加细导出的.在$q=1$的情况下，我们证明了$\ncb (p,1)$是关于逆加细序的格。我们指出，一个类似的发展可以在D型中进行，其中人们得到$\sncd (p,q)$和$\ncd (p,q)$之间的规范同构。对于$q=1$，偏序集$\ncd (p,1)$与Athanasiadis和Reiner在2004年的一篇论文中构造的偏序集“$NC^{(D)} (p+1)$”重合，并且根据该论文的结果是一个格。
英文摘要：
We study the set $\sncb (p,q)$ of annular non-crossing permutations of type B, and we introduce a corresponding set $\ncb (p,q)$ of annular non-crossing partitions of type B, where $p$ and $q$ are two positive integers. We prove that the natural bijection between $\sncb (p,q)$ and $\ncb (p,q)$ is a poset isomorphism, where the partial order on $\sncb (p,q)$ is induced from the hyperoctahedral group $B_{p+q}$, while $\ncb (p,q)$ is partially ordered by reverse refinement. In the case when $q=1$, we prove that $\ncb (p,1)$ is a lattice with respect to reverse refinement order. We point out that an analogous development can be pursued in type D, where one gets a canonical isomorphism between $\sncd (p,q)$ and $\ncd (p,q)$. For $q=1$, the poset $\ncd (p,1)$ coincides with a poset ``$NC^{(D)} (p+1)$'' constructed in a paper by Athanasiadis and Reiner in 2004, and is a lattice by the results of that paper.
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