Reference for Kakutani result on power sum bases of symmetric functions

Question

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?

Answer 1

I think Zagier must have been thinking of the following papers of Kakeya, instead of Kakutani

Kakeya, S.: On fundamental systems of symmetric functions. I, II. Jap. J. Math.2, 69–80 (1925) ; 4, 77–85 (1927)

where the following theorem is established:

Theorem: Suppose that the complement of the sequence $k_1,k_2,\dots,k_n$ is a numerical semigroup, then the power sums $p_{k_1},p_{k_2},\dots,p_{k_n}$ generate the field $\mathbb Q(p_1,p_2,\dots,p_n)$ of symmetric functions in $n$ variables.

He made the conjecture that the converse also holds: if $Q(p_{k_1},p_{k_2},\cdots,p_{k_n})\cong Q(p_1,p_2,\cdots,p_n)$ then $k_1,k_2,\cdots,k_n$ must be the complement of a numerical semigroup. However this is still open in general (for example see section 6 of this recent paper of Dvornicich and Zannier) therefore the two descriptions of sequence A007323 are only conjecturally the same.