How many partial order on a n-set?

Question

Let $A$ be a set has $n$ element, my question is how many partial order on it?

For $n=0,1$, $N_P(n)=1$ Case $n=2$, $N_P(n)=3$ Case $n=3$, $N_P(n)=19$

Is there a general formula?

Update:

It seems closed formula have not been found yet. But does any generating function be known current?

Answer 1

This is the sequence OEIS A001035, the number of partial orders on a labelled $n$-element set; OEIS A000112 gives the number of partial orders on an unlabelled $n$-element set. Neither entry gives a closed formula, a recurrence, or a generating function. Both cite

D. J. Kleitman and B. L. Rothschild, Asymptotic enumeration of partial orders on a finite set, Trans. Amer. Math. Soc., 205 (1975) 205-220

among the references, suggesting that at present all we have are exact results for small $n$ (up to at least $18$ for the labelled case and $16$ for the unlabelled case) and asymptotic results.

Answer 2

A001035 doesn't list a closed-form formula.