Analyzing a function with abstract algebra For an $n$-tuple $S$ of decreasing positive integers, we can define $f(S)$ as subtracting $1$ from every element of $S$, prepending $n$, and then removing $0$s and re-ordering in decreasing order if neccecary. For example, $f((4,2,1))=(3,3,1)$
We are just learning group theory, and our teacher asked us if we can use it to prove that all tuples will eventually reach a cycle when repeatedly applying this function. 
My idea is using $f$ as our sort of group action acting on the set of finite decreasing tuples of positive integers, and then the proof would just be proving all elements have finite order, but I've hit a speedbump - a group action has to be a composition of two elements, i.e $x\cdot y$. This is a function that's just applied to one. I thought of maybe symmetric groups because I heard those have composition as a group action but their elements are functions so I don't think that's helpful here.
So how can this function be analyzed from a group-theoretic perspective? 
EDIT: Accidentally deleted this, I just un-did it. Hope I didn't mess up anything.

You just have to prove that if $x=(x_1,\dots,x_n)$ is your tuple then you only have a finite number of elements in the sequence $$x,f(x),f(f(x)),\dots $$

 Hint: look at the sum of the elements constituting the tuple in $x$ and in $f(x)$.

If you do this then you will necessarily reach twice the same element in the sequence and thus it will lead to a cycle.
Remark: I wouldn't say it is about group theory (maybe about the monoidal action of $\{Id,f,f^2,f^3,\dots\}$). However, I can see why this kind of reasoning is used to show basic group theoretic statements such as: in a finite group $G$, for any $g\in G$, there exists $n_g\in\mathbb{N}$ such that $g^{n_g}=g^{-1}$. 
