When is a function a permutation of the integers?
In his 2011 paper on the Collatz conjecture here Lagarias writes;
> Collatz’s original function, which is a permutation of the integers...
But when is a function over the integers a permutation of the integers? My understanding would be that a permutation must be a bijection from a domain onto itself because only this will reposition every domain element uniquely in the range.
But if we consider the Collatz function exactly a sixth of the integers are mapped to by two distinct elements; namely every even number equivalent to $1\mod 3$, which is is mapped to by both $3x+1$ and $x/2$ e.g.the number 16.
This doesn't seem to be a permutation because $16$ and other numbers will appear multiple times in the range.
Where am I (or Lagarias) going wrong? If it's his mistake, what do you suppose he meant by this?