The number of non-isomorphic arithmetic expressions that can be constructed using +,-,x and /

Abstract

The goal of this paper is to count the number of distinct functions of n variables, up to permutation of the variables, that can be constructed using each variable exactly once, without constants, using only the operations of addition, subtraction, multiplication, and division. We refer to such a function as an arithmetic expression. Under this definition, two expressions are identical if they represent the same rational function; for example, $x_1-x_2-x_3$ and $x_1-(x_2+x_3)$ are identical arithmetic expressions, as are $x_1(x_2+x_3)$ and $(x_2+x_3)x_1$. Two arithmetic expressions are said to be isomorphic if one can be obtained from the other by a permutation of the variables. For example, $(x_1-x_2)/x_3$ and $(x_2-x_3)/x_1$ are isomorphic. The first few values of the number of non-isomorphic arithmetic expressions with n variables are: $$1,4,18,93,500,2844,16621,99674,608448,...$$ In order to accomplish this enumeration, we classify the set of all arithmetic expressions into 12 disjoint categories. Counting all non-isomorphic expressions in each category allows us to obtain the total required quantity.