Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
add comment

2 Answers 2

up vote 2 down vote accepted

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.

share|improve this answer
add comment

A001035 doesn't list a closed-form formula.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.