The number of non-isomorphic equivalence relations on a set of $n$ elements is the partition function $$p(n) =\frac{1}{\pi\sqrt{2}} \sum_{k=1}^{\infty} \sum_{h=1}^{k} \delta_{\gcd(h,k),1} \text{exp}\left(\pi i \sum_{j=1}^{k-1} \frac{j}{k}\left(\frac{hj}{k} - \left\lfloor \frac{hj}{k} \right\rfloor - \frac{1}{2}\right) - \frac{2\pi i h n}{k} \right) \sqrt{k} \frac{d}{dn}\left[ \frac{\sinh\left(\frac{\pi}{k} \sqrt{\frac{2}{3}(n - \frac{1}{24})}\right)}{\sqrt{n - \frac{1}{24}}} \right]$$ The Hardy-Ramanujan asymptotic formula states that $$p(n) \sim \frac{1}{4\sqrt{3}n}e^{\pi \sqrt{2n/3}}$$

By this answer (I would appreciate any reference to an actual derivation of this formula) the number of non-isomorphic relations on a set of $n$ elements is

$$a(n) = \sum_{1s_{1} + 2s_{2} + \cdot\cdot\cdot =n} \left(2^{\sum_{i,j \geq 1} \gcd(i,j)s_{i}s_{j}} \bigg/ \prod_{k=1} k^{s_{k}}s_{k}!\right)$$

I have no idea about the asymptotics of $a(n)$, but if you know of a reference that would be amazing. My question is whether anyone has researched, or if you have any idea about, whether or not $$\frac{p(n)}{a(n)} \sim 0$$ I conjecture that it is asymptotic to zero, but I have no idea how to prove it.

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