Hyperplane arrangements, interval orders, and trees.
Hyperplane arrangements, interval orders, and trees.
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DOI：
10.1073/pnas.93.6.2620
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发表时间：
1996-03
影响因子：
11.1
通讯作者：
R. Stanley
中科院分区：
文献类型：
--
作者：
R. Stanley
超平面排列是实仿射空间中有限个超平面的集合。一个特别重要的排列是辫子排列，它是Rn中所有超平面xi-xj=1，1<；/=i<；j<；/=n的集合。讨论了某些编织变形的组合性质。特别是，与区间序理论和树的计数有意想不到的联系。例如，对可以从一般长度的n个区间I1，…，In中获得的带标签的区间顺序的数目进行计数。还讨论了由于N.Linial的一种排列，其区域数是由Gelfand，Graev和Postnikov[Gelfand，I.M.，Graev，M.I.和Postnikov，A.(1995)，Preprint]定义的交替(或不传递)树的数目。最后，对Shii[Shij-Y.(1986)，《数学讲义》，NO.1179，Springer-Verlag]的一个结果进行了改进，即辫子排列的某个变形有(n+1)n-1个区域，这与按倒数计算标记树有关。
A hyperplane arrangement is a finite set of hyperplanes in a real affine space. An especially important arrangement is the braid arrangement, which is the set of all hyperplanes xi - xj = 1, 1 </= i < j </= n, in Rn. Some combinatorial properties of certain deformations of the braid arrangement are surveyed. In particular, there are unexpected connections with the theory of interval orders and with the enumeration of trees. For instance, the number of labeled interval orders that can be obtained from n intervals I1,..., In of generic lengths is counted. There is also discussed an arrangement due to N. Linial whose number of regions is the number of alternating (or intransitive) trees, as defined by Gelfand, Graev, and Postnikov [Gelfand, I. M., Graev, M. I., and Postnikov, A. (1995), preprint]. Finally, a refinement is given, related to counting labeled trees by number of inversions, of a result of Shi [Shi, J.-Y. (1986), Lecture Notes in Mathematics, no. 1179, Springer-Verlag] that a certain deformation of the braid arrangement has (n + 1)n-1 regions.