A393077 Number of non-isomorphic arithmetic expressions with n variables that can be constructed using +,-,*,/ together with parentheses.

1, 4, 18, 93, 500, 2844, 16621, 99674, 608448, 3770744, 23653630, 149925328, 958737739, 6178529510, 40089044100, 261693178976, 1717542967251

Offset

1,2

Comments

By an arithmetic expression we mean a function consisting of n variables which can be defined by means of the basic binary arithmetic operators +,-,*,/ together with parentheses, where we are allowed to use only once each of the n variables. Under this definition, we distinguish between the syntactic form of an expression and the semantic function it represents; two expressions are considered identical if they represent the same rational function. For example, x-y-z and x-(y+z) are identical arithmetic expressions, as are x*(y+z) and (y+z)*x. Crucial to this definition is that these four operations are used strictly as binary operators. We do not permit unary operations, such as negation or the multiplicative inverse, unless the resulting function can be represented using only binary operators. Two arithmetic expressions are said to be isomorphic if one can be obtained from the other by a permutation of the variables.

Links

Table of n, a(n) for n=1..17.

Boaz Cohen, The number of non-isomorphic arithmetic expressions that can be constructed using +, -, * and /, arXiv:2602.18522 [math.CO], 2026. See references.

Example

For n=3, there are 18 non-isomorphic arithmetic expressions with 3 variables: x+y+z, x-y-z, x+y-z, x*y*z, x*y/z, x/(y*z), (x+y)*z, (x-y)*z, (x+y)/z, (x-y)/z, x/(y+z), x/(y-z), x*y+z, x*y-z, x/y+z, x/y-z, x-y/z, x-y*z.

Crossrefs

Cf. A000084, A140606.

Sequence in context: A306153 A123589 A245103 * A200717 A346763 A121584

Adjacent sequences: A393074 A393075 A393076 * A393078 A393079 A393080

Keyword

nonn,more

Author

Boaz Cohen, Feb 01 2026

Status

approved