TITLE: Finding all posets on a set
QUESTION [8 upvotes]: Suppose I have a set $A$ such that $A$ = $\{1, 2, 3, 4, 5\}$ (or $A$ = $\{1, 2, 3, 4\}$ or $A$ = $\{1, 2, 3\}$ or any other finite small set). How can I find the total number of partial order relations on such a set? 

I'm trying to wrap my head around the concept, but I'm still not quite getting it. (My instructor talked about a Hasse diagram, but am I always to use those/would it make it way easier?) If anyone could explain this concept a bit more clearly than my notes, that would be very good!

REPLY [2 votes]: This is the integer sequence A001035: for n=18 the number you are looking for is 241939392597201176602897820148085023. The problem has been quite extensively studied: see a detailed discussion on the A001035 link above.