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下降多胞形的f向量

The f-vector of the descent polytope

Denis Chebikin, Richard Ehrenborg

备注：
14 pages; to appear in Discrete & Computational Geometry
发表信息：
Discrete & Computational Geometry, Volume 45, Number 3 (2011), 410-424
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摘要：

对于正整数n和[n-1]的子集S，下降多面体DP_S是点x_1，.，x_n在n维单位立方体[0，1]^n中，使得对于S中的i，x_i >= x_{i+1}，否则x_i <= x_{i+1}。首先，我们将DP_S的f-向量表示为[n-1]的所有子集的和。其次，我们使用相关单词在两个字母表上的某些因子分解来描述f向量。我们证明了当集合S是交错集合{1，3，5，.}时f-向量最大化。本文导出了DP_S的f-多项式F_S（t）的母函数，该母函数表示为两个非交换变量的形式幂级数，系数在Z[t]中。我们还得到了下降多面体的Ehrhart多项式的生成函数。

英文摘要：

For a positive integer n and a subset S of [n-1], the descent polytope DP_S is the set of points x_1, ..., x_n in the n-dimensional unit cube [0,1]^n such that x_i >= x_{i+1} for i in S and x_i <= x_{i+1} otherwise. First, we express the f-vector of DP_S as a sum over all subsets of [n-1]. Second, we use certain factorizations of the associated word over a two-letter alphabet to describe the f-vector. We show that the f-vector is maximized when the set S is the alternating set {1,3,5, ...}. We derive a generating function for the f-polynomial F_S(t) of DP_S, written as a formal power series in two non-commuting variables with coefficients in Z[t]. We also obtain the generating function for the Ehrhart polynomials of the descent polytopes.

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