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](https://oeis.org/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?



Reference for Kakutani result on power sum bases of symmetric functions
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](https://doi.org/10.4099/jjm1924.2.0_69), [II](https://doi.org/10.4099/jjm1924.4.0_77). [Jap. J. Math.**2**, 69–80 (1925)](https://doi.org/10.4099/jjm1924.2.0_69) ; [**4**, 77–85 (1927)](https://doi.org/10.4099/jjm1924.4.0_77)
> 
> 
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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](https://doi.org/10.1016/j.aim.2009.06.021) recent paper of Dvornicich and Zannier) therefore the two descriptions of sequence A007323 are only conjecturally the same.


