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Bijections of the natural numbers are equinumerous with functions $\mathbb{N} \to \mathbb{N}$, which are equinumerous with continued fraction expansions of positive irrational numbers.
Edit: Here's one proof of the first assertion. Clearly there are at most as many bijections $\mathbb{N} \to \mathbb{N}$ as functions $\mathbb{N} \to \mathbb{N}$. On the other hand, for any function $f : \mathbb{N} \to \mathbb{N}$ there exists a bijection with $f(i)$ cycles of length $i$. Now apply Cantor-Schroeder-Bernstein.
Bijections of the natural numbers are equinumerous with functions $\mathbb{N} \to \mathbb{N}$, which are equinumerous with continued fraction expansions of positive irrational numbers.