Permutation Statistics of Indexed Permutations
Permutation Statistics of Indexed Permutations
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DOI：
10.1006/eujc.1994.1021
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发表时间：
1994-03
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通讯作者：
E. Steingrímsson
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文献类型：
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作者：
E. Steingrímsson
将对称群Ld的元素的下降性、超越性、主要指标、反转指标和Denert统计量的定义推广到指数置换，即Snd：=Zn？身份证，在哪里？是由{1，2，？，d}的置换关于Ld的通常作用的圈积。双射地证明了超越和下降是等分的，并且相应的下降多项式类似于欧拉多项式，被计算为简单多项式的f-欧拉多项式。证明了下降多项式等于单位d-立方体的某一三角剖分的h-多项式(本质上是h-向量)。一个双射证明了这一点，它利用了这样一个事实，即由三角剖分产生的单纯复形的h向量可以通过对复形的炮击来计算。著名的公式？d？0Edxd/d！=secx+tan x，其中Ed是Ld中交替排列的个数，它被用两种不同的方式推广，一种与V.I.Arnold最近在Morse理论上的工作有关。主指数和逆指数在SND上是均匀分布的。同样，统计量对(d，maj)与统计量对(？，den)是等分布的，其中den是Denert的统计量，？是对超越的另一种定义。推广了与单位d-立方体的某一“片”的体积有关的k次排列的个数。
The definitions of descent, excedance, major index, inversion index and Denert's statistics for the elements of the symmetric group Ld are generalized to indexed permutation, i.e. the elements of the group Snd: = Zn ? Ld, where ? is the wreath product with respect to the usual action of Ld by permutation of {1, 2,?, d}.It is shown, bijectively, that excedances and descents are equidistributed, and the corresponding descent polynomial, analogous to the Eulerian polynomial, is computed as the f-eulerian polynomial of a simple polynomial. The descent polynomial is shown to equal the h-polynomial (essentially the h -vector) of a certain triangulation of the unit d-cube. This is proved by a bijection which exploits the fact that the h-vector of the simplicial complex arising from the triangulation can be computed via a shelling of the complex. The famous formula ?d ? 0Edxd /d! = sec x + tan x, where Ed is the number of alternating permutations in Ld, is generalized in two different ways, one relating to recent work of V. I. Arnold on Morse theory. The major index and inversion index are shown to be equidistributed over Snd . Likewise, the pair of statistics (d, maj) is shown to be equidistributed with the pair (?, den), where den is Denert's statistic and ? is an alternative definition of excedance. A result relating the number of permutations with k descents to the volume of a certain 'slice' of the unit d-cube is also generalized.