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.
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.