How many equivalence relations are there on a set $E$?

Question

I was wondering, just out of pure curiosity, if anyone has ever studied the following function defined over a general space: Given $Ω$ an arbitrary universal set and $Ω^{'} \subseteq Ω$ which is finite, define the function $R:Ω^{'} \to \mathbb N$ where $R(E)$ counts all the possible equivalence relations on $E \times E$. Now this function is well-defined, because we have that $R$ takes finite subsets as inputs, and so $R(E) \leq 2^{|E|^2}$ for all $E \subseteq Ω^{'}$.