Cantor pairing function ($\pi (x,y)=\frac{(x+y)(x+y+1)}{2}+y$)

The Cantor pairing function is an explicit bijection between ${\mathbb{N}}^2$ and $\mathbb{N}$, with the natural numbers numbered from zero:If $d=x+y$, there are $1+2+\cdots +d=d(d+1)/2$ pairs on earlier diagonals, and the pair $(x,y)$ is at offset $y$ along diagonal $d$. The formula therefore gives exactly the enumeration above. For example, $\pi (2,3)=5\cdot 6/2+3=18$.

For the inverse at an index $k$, find the unique $d$ satisfying $d(d+1)/2\le k<(d+1)(d+2)/2$, then put $y=k-d(d+1)/2$ and $x=d-y$. At $k=18$, these are $d=5$, $y=3$, and $x=2$. Thus both the encoding and decoding are explicit. The bijection is useful for assigning a single natural number to a pair of indices, and iterating it does the same for fixed-length tuples.