Let p(n) be the number of partial orders on the set {1,...,n}. From the Online Encyclopedia of Integer Sequences, we find that the known values of p(n) are {1,1,3,19,219,4231,130023,6129859,431723379,44511042511,6611065248783,1396281677105899,414864951055853499,171850728381587059351,98484324257128207032183,77567171020440688353049939,83480529785490157813844256579,122152541250295322862941281269151,241939392597201176602897820148085023}.

We see that the units digits of these numbers appear to cycle with a period of length four: 1, 3, 9, 9.

Experiments with other moduli indicate that given a prime modulus m, the sequence cycles with a period of length m-1. If the modulus m is a prime power, then the period appears to be of length phi(m), where phi is Euler's phi-function. For any modulus m, the period appears to be of length the least common multiple (LCM) of the constituent period lengths. For example, if m=12, the period appears to be of length LCM(phi(4),phi(3))=LCM(2,2)=2.

I don't know how to prove this conjecture and I don't see any reference to it. If proved, perhaps this result together with an asymptotic estimate for p(n) could be used to find higher values of p(n).