Posets are A000112 in Sloane.

The asymptotics aren't given there, but are known. See D. J. Kleitman and B. L. Rothschild, The number of finite topologies, Proc. AMS 25 (1970) 276-282. This paper shows that $\log_2 P_n = n^2/4 + o(n^2)$, where $P_n$ is the number of posets on $n$ elements.

The full asymptotic formula is given in Kleitman and Rothschild, Asymptotic enumeration of partial orders on a finite set, Transactions of the American Mathematical Society 205 (1975) 205-220. This paper gives $\log P_n = n^2/4 + 3n/2 + o(\log n)$, and an explicit (but messy) asymptotic formula for $P_n$.

Edited to add: Richard Stanley, in Enumerative Combinatorics volume 1, exercise 3.3(e) (rated [3+]), gives $$P_n \sim C \cdot 2^{n^2/4+3n/2} e^n n^{-n-1}$$ where $C = {2 \over \pi} \sum_{i \ge 0} 2^{-i(i+1)}$; he states this is a simplification of the formula from Kleitman-Rothschild (1975) that I haven't written out here.

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Posets are A000112 in Sloane.

The asymptotics aren't given there, but are known. See D. J. Kleitman and B. L. Rothschild, The number of finite topologies, Proc. AMS 25 (1970) 276-282. This paper shows that $\log_2 P_n = n^2/4 + o(n^2)$, where $P_n$ is the number of posets on $n$ elements.

The full asymptotic formula is given in Kleitman and Rothschild, Asymptotic enumeration of partial orders on a finite set, Transactions of the American Mathematical Society 205 (1975) 205-220. This paper gives $\log P_n = n^2/4 + 3n/2 + o(\log n)$, and an explicit (but messy) asymptotic formula for $P_n$.