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 $T= \{(p\_0,p\_1, \cdots ,p\_{n-1}) \in (\mathbb Z\_{\geq 0})^n: \sum\_{i=0}^{n-1}p\_i=n \,\, and \,\, \sum\_{i=0}^{n-1}i.p\_i \cong 0 \pmod n\}$. Now my question is how to prove that $S$ is normal ?



Normality of an affine semigroup
I worked on this for a bit with Ricky Liu, who came up with this very quick solution:


Take your set T. Suppose $\sum p\_i = kn$, where $k \geq 2$. Create the following set T': let $i$ appear $p\_i$ times. This creates a set $T'$ with at least $2n$ elements (by your first constraint and $k \geq 2$), whose sum is divisible by $n$ by your second constraint.


However, by [Erdos-Ginsberg-Ziv](http://en.wikipedia.org/wiki/Erd%C5%91s-Ginzburg-Ziv_theorem), there's a subset of $n$ elements which add to $n$, which exactly corresponds to your generator, so we're done.


