On a refinement of the generalized Catalan numbers for Weyl groups
On a refinement of the generalized Catalan numbers for Weyl groups
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关于 Weyl 群的广义 Catalan 数的细化
DOI：
10.1090/s0002-9947-04-03548-2
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
Christos A. Athanasiadis
中科院分区：
文献类型：
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作者：
Christos A. Athanasiadis
设Φ为一个不可约的结晶根系，其Weyl群为W，根格为Q， Coxeter数为h，跨越欧几里德空间V， m为正整数。已知，对于a∈Φ， k = 1,2，…，V中的超平面(α,x) = k分割W的基本腔室的区域集合。， m与W对商Q/ (mh + 1) Q的作用的轨道集相等，描述了这两个集之间的双射，以及Φ根偏序集中某些阶理想链集的双射，并证明了它们在这些集上保持一定的自然统计量。这些集合的元素数量及其相应的细化推广了经典的Catalan数和Narayana数，它们出现在m = 1和Φ = A n-1的特殊情况下。
Let Φ be an irreducible crystallographic root system with Weyl group W, coroot lattice Q and Coxeter number h, spanning a Euclidean space V, and let m be a positive integer. It is known that the set of regions into which the fundamental chamber of W is dissected by the hyperplanes in V of the form (α,x) = k for a ∈ Φ and k = 1, 2,..., m is equinumerous to the set of orbits of the action of W on the quotient Q/ (mh + 1) Q. A bijection between these two sets, as well as a bijection to the set of certain chains of order ideals in the root poset of Φ, are described and are shown to preserve certain natural statistics on these sets. The number of elements of these sets and their corresponding refinements generalize the classical Catalan and Narayana numbers, which occur in the special case m = 1 and Φ = A n-1 .