Symmetric Polynomials and Symmetric Mean Inequalities
Symmetric Polynomials and Symmetric Mean Inequalities
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对称多项式和对称平均不等式
DOI：
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发表时间：
2013
影响因子：
0.7
通讯作者：
Clifford D. Smyth
中科院分区：
文献类型：
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作者：
K. Mahlburg;Clifford D. Smyth
证明了对称多项式拟平均的广义算术-几何平均不等式。不等式满足所有积极的，齐次对称多项式，以及一定的家庭的非齐次多项式，这个家庭允许我们证明以下组合结果标记的正方形网格。假设a $n的细胞 每个方格都是独立填充或空的，其中单元格被填充的概率仅取决于其列。我们证明，对于任何$0 leq ell leq n$，每列最多有$ell$填充站点的概率小于或等于每行最多有$ell$填充站点的概率。
We prove generalized arithmetic-geometric mean inequalities for quasi-means arising from symmetric polynomials. The inequalities are satisfied by all positive, homogeneous symmetric polynomials, as well as a certain family of non-homogeneous polynomials; this family allows us to prove the following combinatorial result for marked square grids. Suppose that the cells of a $n imes n$ checkerboard are each independently filled or empty, where the probability that a cell is filled depends only on its column. We prove that for any $0 leq ell leq n$, the probability that each column has at most $ell$ filled sites is less than or equal to the probability that each row has at most $ell$ filled sites.