What is known about the (asymptotic?) behaviour of the number of equivalence classes of functions $n\to n$, where two functions are considered equivalent if they differ by a permutation of $n$, i.e., $f\sim g$ if there is a permutation $\sigma$ of $n$ so that $f=\sigma g\sigma^{-1}$?


I am guessing this a known or elementary thing that I am somehow unable to google. When restricted to bijections, this is the partition function, which has been famously estimated by Hardy and Ramanujan.



How many functions are there from a set to itself, up to isomorphism?
The OEIS page linked by Sam Hopkins also refers to page 308 of Finch, Mathematical Constants (which refers back to the Meir and Moon paper). Finch writes that the generating function
$$
P(x)=\sum\_1^{\infty}P\_nx^n
$$
satisfies
$$
1+P(x)=\prod\_1^{\infty}(1-T(x^k)))^{-1}
$$
where
$$
T(x)=\sum\_1^{\infty}T\_nx^n
$$
is given on page 296 as the generating function for $T\_n$, the number of nonisomorphic rooted trees of order $n$. $T(x)$ satisfies
$$
T(x)=x\exp\left(\sum\_1^{\infty}{T(x^k)\over k}\right),\quad T\_{n+1}={1\over n}\sum\_1^{\infty}\left(\sum\_{d\mid k}dT\_d\right)T\_{n-k+1}
$$
Finch gives the asymptotic
$$
P\_n\sim\eta\_Pn^{-1/2}\alpha^n 
$$
where
$$
\eta\_P={1\over2\pi}\left({2\pi\over\beta}\right)^{1/3}\prod\_2^{\infty}\left(1-T(\alpha^{-i})\right)^{-1}=0.4428767697\dotsc=(1.2241663491\dotsc)(4\pi^2\beta)^{-1/3}
$$
Here $\alpha=2.9557652856\dotsc=(0.3383218568\dotsc)^{-1}$ is the unique positive solution of the equation $T(x^{-1})=1$, and
$$
\beta={1\over\sqrt{2\pi}}\left(1+\sum\_2^{\infty}\alpha^{-k}T'(\alpha^{-k})\right)^{3/2}=0.5349496061\dotsc
$$
$T'$ being the derivative of $T$.


