is there any explicit injective function $f:\mathbb{P}_{2}(\mathbb{N})\longrightarrow \mathbb{N}$, such that $f$ is a polynomial in two variables? Let $\mathbb{P}_{ 2}(\mathbb{N})=\{A\subset \mathbb{N}:|A|= 2\}$, is there any explicit injective function $f:\mathbb{P}_{2}(\mathbb{N})\longrightarrow \mathbb{N}$, such that $f$ is a polynomial in two variables?
I have tried with $f(\{x,y\})= xy+y+x$, however this function is not injective:
$f(\{x,y\})=x+y+xy=f(\{0,x+y+xy\})$

The problem seems to be discussed in the beginning of this post:
Is There An Injective Cubic Polynomial $\mathbb Z^2 \rightarrow \mathbb Z$?
In particular, the function:
$f(x,y) = (x+y)^2 + y$
Is given as an example, along with some discussion of the same problem over integers.
