I came across the following ring $A$, which appears as a Chow ring. I am wondering if it has been studied before; in particular, I am looking for a reference where this object might have been described.


The graded ring $A^n \subset \mathbb{Z}[x\_1,\dots,x\_n]$ is the subring consisting of polynomials $p$ such that
$$p(x\_1,\dots,x\_{n-1},0)=p(x\_1,\dots,x\_{i-1},0,x\_i,\dots,x\_{n-1})$$
for all $i\geq 1$. For $m\geq n$, there is a surjective homomorphism $A^m \to A^n$ given by setting the last $m-n$ variables to $0$. This is an isomorphism in degrees smaller than $n$. We define
$$A = \varprojlim A^n$$
in the category of graded rings. In particular, the degree $k$ part of $A$ is 
$$A\_{(k)}=A^k\_{(k)}.$$



Identify ring of polynomials symmetric under forgetting variables
The ring $A^n$ is the set of polynomials $f\in\mathbb{Z}[x\_1,\ldots,x\_n]$ such that the coefficient of $x\_{i\_1}^{a\_1}\ldots x\_{i\_k}^{a\_k}$ is equal to the coefficient of $x\_{j\_1}^{a\_1}\ldots x\_{j\_k}^{a\_k}$ whenever $i\_1<\ldots<i\_k$ and $j\_1<\ldots<j\_k$. Polynomials satisfying this condition are called quasi-symmetric, and $A$ is the ring of quasi-symmetric functions.


