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I've been reading about lattices and partial orders (my reference: Applied Abstract Algebra by Lidl, Pilz) while this question struck me.

Let X be a finite set. Is there any way to determine the total number of non-isomorphic partial orderings of this set?

I searched math.SE a bit, but couldn't find any similar question. Any kind of response/help will be appreciated. Best regards.

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    @MichaelHardy : Yes, that is what I actually meant. I thought every partial order has a unique representation by Hasse diagram. I've changed the question accordingly. Thanks a million for pointing this out.2012-04-27

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No general result seems to be known; known values beyond the trivial ones seem to have been found by computer enumeration. This is sequence A001035 in the On-Line Encyclopedia of Integer Sequences; the first link has a number of references that may be of interest.

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    @MichaelHardy :Thanks.2012-04-26