# Number of lattices over a finite set

I'm interested in finding a general formula for the number of lattices possible over a finite set $$S$$ as a function of the set's cardinality.

For instance, how many lattices over $$\{1, 2, 3\}$$ are there? Since the set contains three elements, and a lattice is a partial order, we must count the number of ways in which we can order three elements ensuring that the supremum between any pair always exists. There are $$3! = 6$$ total orders which satisfy this constrain, and in this case none of the non-total orders satisfy the constraint. $$\therefore$$ There are $$6$$ possible latices over $$\{1, 2, 3\}$$.

The case with only three elements is simple because only total orders count and there always are $$|S|!$$ total orders. But as non-total orders become available, the problem becomes slightly more difficult. Is there a general form (e.g. a generating function) for the number of lattices over a set $$S$$ of given cardinality?

(Similar questions (e.g. this one) have been asked but none, as far I researched, related to lattices.)

• Commented Jul 12 at 18:31
• Thanks for pointing that out, it is helpful! My question still is about a general formula for that sequence. Commented Jul 12 at 18:47
• @lafinur In the linked page, there is another link to a paper of Heitzig and Reinhold, Counting finite lattices. There they have algorithms to compute both labeled and unlabeled numbers of lattices in a $n$-element set. It seems that each value is computed using all the previous ones, so it's probably exponential time on the size $n$. They show the values up to $n=18$. The paper is kind of technical and the algorithms depend on concepts which cannot be defined in a comment. Commented Jul 12 at 20:28
• @amrsa I appreciate the reference. I did a quick read and evidently there is no closed-form formula. Interesting paper. Thanks. Commented Jul 12 at 21:37

As suggested in the comments, there is no closed-form formula (yet) for the number of lattices over a finite set of $$n$$ elements. See this paper.