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?