Functional composition patterns and power series reversion
Functional composition patterns and power series reversion
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DOI：
10.1090/s0002-9947-1960-0114765-9
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发表时间：
1960-03
影响因子：
1.3
通讯作者：
G. N. Raney
中科院分区：
文献类型：
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作者：
G. N. Raney
见于 Jacobson [9]、Becker [2]、Motzkin [11] 和 Bourbaki [3] 的著作； 4]。本文将关注凯莱问题的自然推广，并将表明推广问题的解决方案包含建立著名的幂级数回归拉格朗日公式所需的所有组合信息。为了描述这个问题，我们考虑使用运算符前缀表示法从运算符符号和参数符号构建的表达式。权重被分配给表达式中的符号，自变量符号的权重为0，n元运算符符号的权重为n。表达式有多种类型，表达式的类型仅取决于其中符号的权重以及它们出现的顺序。例如，表达式 (a+b) +c 写作 + +abc，类型为 22000，而 a+(b+c) 写作 +a+bc，类型为 20200。表达式 F(G(x, H(y, z), t), K(u)) 类型为 230200010。按照 P. C. Rosenbloom [13]，我们将这些有限序列称为自然数的有限序列。指定表达式“单词”类型的数字。这些序列的定义和一些特殊属性在第 2 节中说明。
found in the writings of Jacobson [9], Becker [2], Motzkin [11], and Bourbaki [3; 4]. This paper will be concerned with a natural generalization of Cayley's problem, and will show that the solution to the generalized problem contains all of the combinatorial information needed to establish the well known formula of Lagrange for the reversion of power series. To describe the problem, we consider expressions which are built from operator symbols and argument symbols, using a prefix notation for operators. Weights are assigned to the symbols in an expression, an argument symbol having the weight 0 and an n-ary operator symbol having the weight n. Expressions are of various types, the type of an expression depending only on the weights of the symbols in it and on the order in which they appear. The expression (a+b) +c, for example, is written + +abc and is of the type 22000, while a+(b+c) is written +a+bc and is of the type 20200. The expression F(G(x, H(y, z), t), K(u)) is of the type 230200010. Following P. C. Rosenbloom [13], we call those finite sequences of natural numbers which designate the types of expressions "words." Definitions and some special properties of these sequences are stated in ?2.