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The minimal number of invariants needed to generate ${\mathbb Z}[x_1,...,x_n]^{{\mathbb Z}}_n}$ Z}_n}$has been considered by a number of authors including Erdos, Dixmier and Kac. (see the references in [John C. Harris and David L. Wehlau Non-Negative Integer Linear Congruences, Indagationes Mathematicae 17 No. 1 (2006) 37-44]. It is easily seen to be bounded below by the number of partitions of n,${\mathcal P}(n)$. Dixmier produced a number of papers giving the asymptotic behavior of this number as a function of$n$. The results in the above Harris-Wehlau paper are completed by using the main result in [Pingzhi Yuan, On the index of minimal zero-sum sequences over finite cyclic groups, Journal of Combinatorial Theory, Series A 114 (2007) 1545–1551]. These two papers combine to show that the number of homogeneous generators of this ring of invariants of degree$k$is exactly$\phi(n){\mathcal P}(n-k)$if$k \geq \lfloor n/2\rfloor + 2$(here$\phi$is Euler's totient function). Surprisingly (at least to me) much less is known about the number of generators in lower degrees. 3 corrected spelling error The minimal number of invariants needed to generate${\mathbb Z}[x_1,...,x_n]^{{\mathbb Z}}_n}$has been considered by a number of authors including Erdos, Dixmier and Kac. (see the references in [John C. Harris and David L. Wehlau Non-Negative Integer Linear Congruences, Indagationes Mathematicae 17 No. 1 (2006) 37-44]. It is easily seen to be bounded below by the number of partitions of n,${\mathcal P}(n)$. Dixmier produced a number of papers giving the asymptotic behavior of this number as a function of$n$. The results in the above Harris-Wehlau paper are completed by using the main result in [Pingzhi Yuan, On the index of minimal zero-sum sequences over finite cyclic groups, Journal of Combinatorial Theory, Series A 114 (2007) 1545–1551]. These two papers combine to show that the number of homogeneous generators of this ring of invariants of degree$k$is exactly$\phi(n){\mathcal P}(n-k)$if$k \geq \lfloor n/2\rfloor + 2$(here$\phi$is Eulor's Euler's totient function). Surprisingly (at least to me) much less is known about the number of generators in lower degrees. 2 added 12 characters in body The minimal number of invariants needed to generate${\mathbb Z}[x_1,...,x_n]^{{\mathbb Z}}_n}$has been considered by a number of authors including Erdos, Dixmier and Kac. (see the references in [John C. Harris and David L. Wehlau Non-Negative Integer Linear Congruences, Indagationes Mathematicae 17 No. 1 (2006) 37-44]. It is easily seen to be bounded below by the number of partitions of n,${\mathcal P}(n)$. Dixmier produced a number of papers giving the asymptotic behavior of this number as a function of$n$. The results in the above Harris-Wehlau paper are completed by using the main result in [Pingzhi Yuan, On the index of minimal zero-sum sequences over finite cyclic groups, Journal of Combinatorial Theory, Series A 114 (2007) 1545–1551]. These two papers combine to show that the number of homogeneous generators of this ring of invariants of degree$k$is exactly$\phi(n){\mathcal P}(n-k)$if$k \geq \lfloor n/2\rfloor + 2$(here$\phi\$ is Eulor's totient function). Surprisingly (at least to me) much less is known about the number of generators in lower degrees.