I've run across some polynomials that are natural enough I imagine must have been named and studied, but don't know what they're called.
In the ring $\mathbb{Z}[x_i,y_i : i \in I]$ one has polynomials \begin{align*} p_{1,0} &= \sum x_i,\\ p_{2,0} &= \sum_{i \neq j} x_i x_j,\\ &... \\ p_{0,1} &= \sum y_i,\\ p_{0,2} &= \sum_{i \neq j} y_i y_j\\ &... \\ \end{align*} which are elementary symmetric variables in one set of variables or the other, but I also have sort of relatives based on partitions, e.g., \begin{align*} p_{1,1} &= \sum_{i \neq j} x_i y_j,\\ p_{2,1} &=\sum_{|\{i,j,k\}| = 3} x_i x_j y_k, \end{align*} and so on: in $p_{m,n}$ one sums all products of $m$ of the $x$ variables and $n$ of the $y$ variables with all indices distinct. Something similar can of course be done to define polynomials $p_{m,n,r}$ in $\mathbb Z[x_i,y_i,z_i]$ and so on.
What are they called?
Look at the Hopf algebra structure on symmetric functions.
This is given by sending $f(x_1 , \cdots )$ to $f(x_1 , x_2 \cdots y_1 , y_2 \cdots )$ by any bijection $(x_1 ,x_2 \cdots ) \to (x_1 , x_2 \cdots y_1 , y_2 \cdots )$. On the elementary $e_j$, you get $\Delta e_n = \sum_{i+j=n} e_i \otimes e_j$