  A bijection $(l,r)$ between $M^2$ and $M$ is said to be a pairing function
with no cycles, if any composition of its coordinate functions has no fixed
point. We compute here the Grothendieck ring of the pairing function without
cycles to be isomorphic to $\mathbb{Z}^2\simeq \mathbb{Z}[X]/(X-X^2)$. More
generally, for any $n\in \mathbb{N}^*$ and any bijetion without cycles betwen
$M$ and $M^n$, the exact same method proves that
$K_0(M)=\mathbb{Z}[X]/(X-X^n)$.
