# Listing all posets on 9 points?

I'm looking for a list of all (non-isomorphic) posets on 9 points. I know there are 183231 of them (OEIS A000112), but in order to progress with a problem I'm working on, I'd need the posets themselves, not only their number.

Background: A poset is a set $$S$$ together with a relation $$\le$$ on $$S$$ which is reflexive, transitive, and antisymmetric. Elements of $$S$$ are called points here. Two posets are isomorphic if there is a relation-respecting bijection between them. Deciding whether two posets are isomorphic is known to be a difficult problem, and the numbers of isomorphism classes of finite posets are known only for posets with up to 16 points (see the OEIS entry cited above). State of the art is a 2002 paper by Brinkmann and McKay: "Posets on up to 16 Points", Order 19 (2) (2002) 147-179.

• John Stembridge has been so kind to provide me with a list of all posets on 8 points, which has been of great help to solve my problem for 8-point structures. Now I'd like to one-up my research, but without access to all 9-point posets, that will be hopeless. – Uli Fahrenberg Feb 25 at 23:13

See the "Partially-ordered sets (posets)" section here, where Brendan McKay has provided the posets up to 10 points.

• I used this data to create a Mathematica package, to convert from his representation to the 'all-edges' representation: github.com/PerAlexandersson/Mathematica-packages/blob/master/… Unfortunately, I have not included the size 9 posets (due to size, I think), but one can more or less copy-paste that data there. In any case, people who like Mathematica might find this useful – Per Alexandersson Feb 26 at 9:08

Using posets(9) from Finite posets in SageMath will give you an iterator of all posets (up to isomorphism) on 9 elements. So, this isn't a list you can download like in the other answer, but gives a way to work with posets on $$n$$ elements up to isomorphism. I tried on my laptop and was able to iterate on 10% of the posets on 9 elements in about 4 minutes.

The following file lists adjacency matrices of all 183231 inequivalent partial orders (i.e., $$T_0$$ topologies) on 9 points, with spaces separating rows:

https://www.mathtransit.com/finite_topological_spaces/9-spaces_T_0_binary.txt.gz

I created this file from one that was at the now-defunct Chapel Hill Poset Atlas, the front page of which can still be viewed here:

http://web.archive.org/web/20190905161049/https://lists-of-posets.math.unc.edu/

In late 2019 I posted files containing all nonhomeomorphic topologies on spaces up to and including 9 points, here:

https://www.mathtransit.com/finite_topological_spaces.php

I am currently in the process of expanding this page to include files for cardinalities 10 and 11, the latter of which will probably take the form of a simple cookbook recipe due to its large size.