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According to Birkhoff in 'Lattice theory' there are 19 3-element posets. In Sierpinski, 'Cardinal and ordinal numbers', 17 of them are exhibited. But Sierpinski seems to agree with Birkhoff (in as much as he quotes him on p. 192 which is where I got the number 19 from, I have not seen Birkhoff) and he says nothing about the 19 vs. 17 discrepancy. Anyway, that leads me a the obvious question: is there an easy way to calculate how many posets there are on an $n$-element set? Sierpinski says no, but that was in 1958 and maybe more is known now!

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    $\begingroup$ oeis.org/A001035 has some data, but as for a formula it refers to other OEISs. Being a simple-minded fellow, I like simple formulae! $\endgroup$ – Justin Aug 10 '19 at 9:36

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