Number of isomorphism types of functions f:[n]->[n] Consider the set $F_n: = [n]^{[n]}$ of all functions $f:[n] \rightarrow [n]$, $[n] = \lbrace 1,2,...,n\rbrace$. It is well known that $|F_n| = n^n$. 
Edit: Let two functions $f, g$ in $F_n$ be of the same isomorphism type ($f\sim g$) iff there exists a permutation $\pi$ such that $f\pi = \pi g $

What is the number of isomorphism types of functions $f:[n] \rightarrow [n]$, i.e. what is $|F_n/_{\sim}|$?

Examples:


*

*$|F_2| = 4$, $|F_2/_{\sim}| = 3$ 

*$|F_3| = 27$, $|F_3/_{\sim}| = 7$


If you want to count the number of functions up to renamings on the domain and codomain, then the number is $p_n$, the number of partitions of $n$.
Later: now that you've made precise what you wanted... This is counted here, with references.
