论文标题
B类中的Q-Stirling号码
q-Stirling numbers in type B
论文作者
论文摘要
在对称组中计算集合和排列的分区的Stirling数字已在组合，几何和代数中发现了广泛的应用。我们研究了与B型Coxeter相对应的这些数字的类似物和Q分析，尤其是我们展示了它们与完整的均质和基本对称多项式相关的关系。演示他们如何Q计数签名的分区和排列；计算其普通，指数和Q指数生成函数；并证明它们的各种身份。最近出现了第二种Q-striring数字的订购类似物，这是Zabrocki和Swanson的猜想 - 关于某些超级共同变量代数的希尔伯特系列的Wallach。我们为这些代数提供了猜想的基础，并表明它们具有正确的希尔伯特系列。
Stirling numbers, which count partitions of a set and permutations in the symmetric group, have found extensive application in combinatorics, geometry, and algebra. We study analogues and q-analogues of these numbers corresponding to the Coxeter group of type B. In particular, we show how they are related to complete homogeneous and elementary symmetric polynomials; demonstrate how they q-count signed partitions and permutations; compute their ordinary, exponential, and q-exponential generating functions; and prove various identities about them. Ordered analogues of the q-Stirling numbers of the second kind have recently appeared in conjectures of Zabrocki and of Swanson--Wallach concerning the Hilbert series of certain super coinvariant algebras. We provide conjectural bases for these algebras and show that they have the correct Hilbert series.
