Baxter algebras and combinatorial identities. II
Baxter algebras and combinatorial identities. II
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DOI：
10.1090/s0002-9904-1969-12156-7
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发表时间：
1969-03
影响因子：
1.3
通讯作者：
G. Rota
中科院分区：
文献类型：
--
作者：
G. Rota
1.导论.壮观的结果波动理论的总和独立随机变量，在过去15年中获得的安德森，巴克斯特，Bohnenblust，福阿塔，肯珀曼，斯皮策，Takacs和其他人，逐步导致实现的性质问题，以及解决办法，是代数和组合。后巴克斯特表明，关键的问题在于简化某一经营者身份，几个代数证明（阿特金森，金曼，温德尔）。这是目前的目的进行这种代数化的限制：我们目前的结果相当于一个解决方案的字问题的巴克斯特代数。该解决方案不是作为一个算法，但通过显示，每个身份在巴克斯特代数是有效地等价于一个身份的对称函数的变量的数量无关。值得注意的是，迄今为止使用的身份在波动理论的组合学“翻译”本方法成经典的对称函数身份容易验证。然而，本方法对于猜测和证明新的组合恒等式也是有用的：通过示例，它将在本说明的第二部分中显示它如何导致Bohnenblust-Spitzer公式对于任意排列群的作用的推广。一个类似的不等式理论将在其他地方提出。
1. Introduction. The spectacular results in the fluctuation theory of sums of independent random variables, obtained in the last 15 years by Andersen, Baxter, Bohnenblust, Foata, Kemperman, Spitzer, Takacs and others, have gradually led to the realization that the nature of the problem, as well as that of the methods of solution, is algebraic and combinatorial. After Baxter showed that the crux of the problem lay in simplifying a certain operator identity, several algebraic proofs (Atkinson, Kingman, Wendel) followed. It is the present purpose to carry this algebraization to the limit: the result we present amounts to a solution of the word problem for Baxter algebras. The solution is not presented as an algorithm, but by showing that every identity in a Baxter algebra is effectively equivalent to an identity of symmetric functions independent of the number of variables. Remarkably, the identities used so far in the combinatorics of fluctuation theory" translate" by the present method into classical symmetric function identities of easy verification. The present method is nevertheless also useful for guessing and proving new combinatorial identities: by way of example, it will be shown in the second part of this note how it leads to a generalization of the Bohnenblust-Spitzer formula for the action of arbitrary groups of permutations. A parallel theory of inequalities will be presented elsewhere.