A partially ordered set (poset) is a set of points where some points sit above others, consistently. Up to relabeling, there are 378,180,373,015,320,248 posets on 17 points. The count had stopped at 16 points since 2002. This number comes from the labeled counts, which are known further, by averaging over all permutations of the points. Almost every permutation's contribution is a labeled count that is already known; the rest come from a few exact statistics.
Burnside's lemma: the number of posets up to relabeling equals the average, over all N! permutations g of the points, of the number of labeled posets that g leaves unchanged. Permutations with the same cycle type leave the same number unchanged, so the sum runs over cycle types.
A poset that g leaves unchanged has a simple shape. Each cycle of g is a set of mutually incomparable points, the cycles themselves form a smaller poset (the quotient), and between two comparable cycles the poset only records a set of offsets. When the cycle lengths have no common factors, every offset set is forced, so the count is just the labeled count on the cycles. Pick a cycle type below to see it worked out.
| cycle type | permutations | kind | fixed posets | contribution |
|---|
Hasse diagrams: each point is drawn above the points it covers. Solid cards are connected (A000608); dashed cards split into pieces. The number under each diagram is its symmetry count, and n!/symmetries is how many labeled posets it stands for.