# What is the number of maximal antichain in a poset?

This is a topic I am recently working on. Given a poset, how many different antichains are there? I find little literature on it. And I am interested whether there is a closed formula, or a tight lowerbound or upperbound given the structure of the poset, or approximation ratio guaranteed algorithm.

There is a post concerning "number of antichain in poset" herelink text.

Thank you very much in advance. :-)

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So, what is the given structure of the poset? The answer will depend heavily on it. – Emil Jeřábek Aug 3 '11 at 14:28
at that link you can find how to compute an upperbound. Moreover, if m is the size of the maximal antichain, then 2^m is a lower bound (and of course it can be refined, but I don't know how much...) But if you get more interesting answers, I'm still interested in this question! – klaraspina Aug 3 '11 at 15:04
hi Emil, what if it is just a poset? – klaraspina Aug 3 '11 at 15:05
What I meant that more information about properties of the poset is needed to get a sensible answer. Otherwise, the number of maximal antichains may be on the one hand as small as $1$ (for a discrete poset) and on the other hand almost exponential in the size of the poset (for a Boolean algebra). – Emil Jeřábek Aug 3 '11 at 15:23
I have never understood what people mean when they say "closed formula" in this kind of generality. Closed formula in terms of what? – Qiaochu Yuan Aug 3 '11 at 15:52