From "Models and Games" by Jouko Vaananen (Cambridge studies in advanced mathematics), I quote



> 
> The Hardy-Ramanujan asymptotic formula says that the number of equivalence relations on a fixed set of $n$ elements is asymptotically
>  $$\frac{1}{4\sqrt{3}n}e^{\pi\sqrt{2n/3}}$$
>  So this is also an asymptotic upper bound for the number of non-isomorphic equivalence relations on a universe of $n$ elements.
> 
> 
> 


I am very confused by this as I thought that the Hardy-Ramanujan asymptotic formula was asymptotic to the partition function $p(n)$ which says how many times you can write $n$ as a sum of positive integers, not the number of partitions of a set of $n$ elements. Is this book incorrect or am I missing something about the connection between equivalence relations and the partition function?



Does the Hardy-Ramanujan Asymptotic Formula Partition Sets or Integers?
If $R$ and $S$ are two equivalence relations on a set $X$ of card. $n$, we say that they are equivalent if there exists a bijection $\varphi:X\rightarrow X$ such that:


$\varphi(x)R\varphi(y)\quad\iff\quad xSy$


Then the partition function $n$ is counting the number of equivalence relations on $X$ modulo this identification.


