Functional composition patterns and power series reversion

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.