Numerical semigroups are additive submonoids $A$ of the natural numbers such that the greatest common divisor of all elements of $A$ is 1. The complement of a numerical semigroup in $\mathbb{N}$ is finite and is called the genus of the numerical semigroup.

The sequence A007323 in the OEIS gives the number of numerical semigroups with a given genus $g$. Zagier writes in the comments that this also gives the number of sets of $g$ power sum symmetric functions which form a $\mathbb{Q}$-algebra basis for the field of symmetric functions in $g$ variables. He attributes this result to Kakutani but says he forgot the reference.

Does anyone know the source of this result?