An affine monoid is a finitely generated commutative submonoid of $\mathbb Z^k$ for some positive integer k. Let S be an affine monoid and let G(S) be the group generated by S. We say the monoid S is normal if and only if for all $g \in G(S)$ and $n \in \mathbb N \setminus {0}$, $ng \in S$ implies $g \in S$.

Let $S$ be the submonoid generated by the finite set $T= {(p_0,p_1, \{(p_0,p_1, \cdots ,p_{n-1}) \in (\mathbb Z_{\geq 0})^n: \sum_{i=0}^{n-1}p_i=n \,\, and \,\, \sum_{i=0}^{n-1}i.p_i \cong 0 mod (n)}$. \pmod n\}$. Now my question is how to prove that $S$ is normal ?

An affine monoid is a finitely generated commutative submonoid of $\mathbb Z^k$ for some positive integer k. Let S be an affine monoid and let G(S) be the group generated by S. We say the monoid S is normal if and only if for all $g \in G(S)$ and $n \in \mathbb N \setminus {0}$, $ng \in S$ implies $g \in S$.

Let $S$ be the submonoid generated by the finite set $T= {(p_0,p_1, \cdots ,p_{n-1}) \in (\mathbb Z_{\geq 0})^n: \sum_{i=0}^{n-1}p_i=n \,\, and \,\, \sum_{i=0}^{n-1}i.p_i \cong 0 mod (n)}$. Now my question is how to prove that $S$ is normal ?