Count all the function f satisfying followings :

Question

For $A=\{1,2,3,4,5,6,7,8,9,10\}$

Define a function $f : A\to A.$

Then 30 times composite of f, that is ; $f\circ f\circ...\circ f(x) = x$ and 30 is the least number for f to become an identity.

How many $f : A\to A$ are there ?

Answer 1

First: from $f \circ \cdots \circ f (x) = x$, are you able to conclude that $f$ is bijective (therefore a "permutation" of $A$)?

Second: Are you familiar with the decomposition of permutations into a composite of disjoint cycles?

Third: If you are, do you know the equation for the order of a permutation (the smallest "power" of that permutation, greater or equal to one, that equals the identity permutation), as a function of the lengths of the cycles in a disjoint cycle decomposition?

Fourth: if you can answer the other questions, can you now "count" to get to the answer?

This will have something to do with the fact that the only way you can write 10 as the sum of one or more strictly positive integers whose least common multiple is 30 is as $2+3+5$.

If you have difficulty with any of these steps please write back and we can elaborate. Good luck!