Perhaps there are none with integral coefficients; so let us admit rational coefficients. The map $(x, y) \mapsto x + \frac{1}{2}(x + y)(x + y + 1)$ is well known, and swapping $x$ and $y$ in the formula yields another, so we have two for starters.



What polynomials biject from $\mathbb{N}^{2}$ to $\mathbb{N}$?
Describing such bijections is an open problem. Maximal result (there is no other bijections among polynomials of degree not higher than 4) are contained in



> 
> John S. Lew, Arnold L. Rosenberg,
>  *Polynomial indexing of integer lattice-points I. General concepts and quadratic polynomials*, J. Number Theory **10** (1978) pp 192-214, doi:[10.1016/0022-314X(78)90035-5](http://dx.doi.org/10.1016/0022-314X%2878%2990035-5).  
> 
> *Polynomial indexing of integer lattice-points II. Nonexistence results for higher-degree polynomials*, J. Number Theory **10** (1978) pp 215-243,
>  doi:[10.1016/0022-314X(78)90036-7](http://dx.doi.org/10.1016/0022-314X%2878%2990036-7)
> 
> 
> 


