# [seqfan] Re: Planar distributive lattices

David Seal david.j.seal at gwynmop.com
Wed Apr 21 12:24:44 CEST 2021

```With regard to question (a), I think the Wikipedia pages https://en.wikipedia.org/wiki/Distributive_lattice and https://en.wikipedia.org/wiki/Hasse_diagram contain the required information. My only uncertainty is whether 'planar' is being used as shorthand for 'that has an upward planar Hasse diagram', but the poster clearly shows Hasse diagrams that are upward planar, and the diagrams do not include one of the subsets of a 3-element set {x,y,z} ordered by subset inclusion. That lattice has a Hasse diagram which is the skeleton of a cube (see https://en.wikipedia.org/wiki/Partially_ordered_set#/media/File:Hasse_diagram_of_powerset_of_3.svg), and the skeleton of a cube is of course a planar graph - but drawing it in a planar fashion necessarily causes it to violate the 'upward' property of Hasse diagrams and so it isn't upward planar. So the fact that the poster's diagrams do not include the skeleton of a cube tends to support my belief that in 'planar distributive lattice', 'planar' is referring to upward planarity of the Hasse diagram and not just planarity of a diagram.

With regard to question (c), I've checked Dr. Jipsen's poster for numbering the diagrams correctly and the diagrams being listed in non-decreasing order of their vertex count (so that all listed diagrams with 1 vertex come before all listed diagrams with 2 vertices, which come before all listed diagrams with 3 vertices, etc). That means that the sequence can be determined by a(n) = (number of first diagram with n+1 vertices) - (number of first diagram with n vertices). Doing that does indeed confirm your counts.

With regard to question (d), the decision seems to have already been made to include it in the OEIS - and for what little it's worth, I completely agree with that decision.

With regard to question (b), I'm afraid I cannot currently confirm whether Dr. Jipson has enumerated them correctly, but I didn't notice any obvious problems such as duplicates or clear omissions.

David

> On 15/04/2021 02:08 Allan Wechsler <acwacw at gmail.com> wrote:
>
>
> I forget how I stumbled on this:
> https://math.chapman.edu/~jipsen/mathposters/Planar%20distributive%20lattices%20up%20to%20size%2011.pdf
> .
>
> It is a chart purporting to show all of the planar distributive lattices
> with up to 11 vertices. Like any true-hearted sequence fanatic I counted
> the number of these guys of each order, and got the following sequence:
>
> 1,1,1,2,3,5,8,14,24,42,72...
>
> Imagine my surprise at finding this sequence missing from OEIS! The author
> is apparently Dr. Peter Jipsen, at Chapman University in California.
>
> Perhaps someone here can figure out (a) what a planar distributive lattice
> is, (b) whether Dr. Jipsen enumerated them correctly, (c) whether I counted
> them off Jipsen's poster correctly, and (d) whether to add the sequence.
>
> Thank you!
>
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