Roots of maps of finite sets Let $X_n$ be a set with $n$ elements.  Write $F(X_n,X_n)$ for the set of maps from $X_n$ to itself.  We give it the operation of composition. I am curious if there is a nice formula for the following question.  Let $m$ be a positive integer.   How many maps in $F(X_n,X_n)$ are $m$th powers of other maps?  In other words, how big is the image of the function which takes each map to its $m$th power?  This is just for fun.  Thanks!
 A: As I mentioned in comments, what you denote $F(X_n,X_n)$ is known as the full transformation semigroup on $X_n$,  often denoted by $\mathcal{T}_{X_n}$, or simply $\mathcal{T}_n$. 
One reference I found is Digraphs and the semigroup of all functions on a finite set, by Peter M. Higgins, Glasgow J. Math. 30 (1988), pp. 41-57; available here. There he considers, inter alia, the problem of finding all solutions to $x^m = c$ in the full transformation semigroup. 
Having the standard name for the object you are looking at may faciliate finding references.
Addendum. Another reference handles the case $m=2$, and provides a characterization of exactly when an element of the full transformation semigroup is a square; this is a 1982 paper of Howie and Snowden, Square roots in finite full transformation semigroups, Glasgow J. Math. 23 (1982), 137-149 (available here). As the authors note, the characterization is "disappointingly complicated", and I don't know how amenable it is to trying to count the number of squares in $\mathcal{T}_n$. It would seem that this is actually a fairly difficult problem, though of course a lot of progresss may have been made in the past 20-30 years.
