A current project uses bijections from a set to itself. (The set is the integer compositions of $n$, i.e., "ordered partitions of $n$," but that doesn't seem pertinent to the question.) Is there a more specific name for such maps? These do not have order two, so involution is not correct. There is not an algebraic structure being considered, so automorphism doesn't sound right...



Terminology for a bijection from a set to itself
Permutation is the term I would use (indeed, when I teach, I define a "permutation" of a set $X$ as a bijection from $X$ to itself).


