A problem for convex set = disk polyominoes, more terms

[brad klee]

Isn't A147680 a subset of A181785? 

These should be related more explicitly through cross refs, as A181785 seems to be 

disk polyominoes when the disk is any convex subset of E euclidean plane: 

https://en.wikipedia.org/wiki/Convex_set

The values of A181785 should be easy to extend now. Harm.On.ica wrote two 

different algorithms and got the same values out to a(25): 

https://github.com/bradleyklee/Artifacts/tree/main/14-convex-polyominoes

1,1,2,5,10,25,48,107,193,365,621,1082,1715,2777,4247,6519,9608,

   14210,20263,28957,40411,56307,76929,104907,140431
 

We have to be careful though because thin disks even with quadratic boundary 

functions can included totally disconnected unit squares (picture below). 

Perhaps simple connected through edges is assumed as a requirement in any 

definition by intersection of continuous and discrete sets. 

 0.00,

--brad

[Arthur O'Dwyer]

On Mon, Jun 22, 2026 at 3:23 PM 'brad klee' via SeqFan <seq...@googlegroups.com> wrote:

Isn't A147680 a subset of A181785? 

It's true that A181875(n) >= A147680(n) for all n, because all disk polyominoes are convex polyominoes. I'll submit a comment to that effect.

These should be related more explicitly through cross refs, as A181785 seems to be 

disk polyominoes when the disk is any convex subset of E euclidean plane:

A "disk" is a circle plus its interior.

It's the 2D analog of a 3D "ball," which is a sphere plus its interior.

 

[...]

We have to be careful though because thin disks even with quadratic boundary 

functions can included totally disconnected unit squares (picture below). 

In other words, "Is

X..

..X

a convex polyomino?" The answer is no; it's not any kind of polyomino. Polyominoes are connected by definition.

–Arthur

[Allan Wechsler]

To elaborate on Arthur's comment about "convex polyominoes"; the definition at oeis.org/A181785 is careful to say that we start with a polyomino (viewed, again, as a collection of lattice-points, not as a set of square tiles), and then judge whether it's convex by constructing its convex hull and intersecting the resulting polygon with Z^2. So the restriction to connectivity is baked into the definition; if this is at all ambiguous it should be clarified in the entry.

By all means there should be cross-reference comments. I still have draft edits at oeis.org/A147680, which should not be submitted for review until I finish writing enumeration code that we have confidence in, and until the data is thereby corrected.

-- Allan

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[brad klee]

Thanks. Yea, I think the proof is kind of obvious. 

Assume point set X is circular disk but not convex disk. The convex hull lies entirely inside the 

circular boundary and it must contain another point we haven't included yet. By nestedness 

of convex and circular hulls, the same point must be found within the circular boundary,

thus we obtain a contradiction. We have no similar problem when X is Circular disk and 

Convex disk, but we can find points outside the convex hull and inside the circular hull. 

From the existence and properties of elliptical convex disks, we have intuition that the 

circular set must be asymptotically a much smaller (<<) set than the convex set.

It's probably also easy to prove that limit_{n->infty} A147680(n) / A181785(n) = 0, just 

by considering the circle as a special case of the ellipse. Or possibly by introducing 

circular harmonics. 

As for your assertion of circular disks and where to find them, that can be your story. 

It's the normal story, but that's not my story. 

My preference is for Convex Disks, and I will point out that people have already made

hexagonal shaped discs to play disk golf with (videos on youtube, they fly just fine).   

We don't have to have the same story until the style guide changes, in which case, you could 

potentially win the more standard definitions contest. That depends what's already written down 

in pre-existing entries. 

Inductively speaking: How do pre-existing entries use the "disk" term? 

If anyone agrees A181875 deserves more terms added, here's again what I got:

https://github.com/bradleyklee/Artifacts/tree/main/14-convex-polyominoes

1,1,2,5,10,25,48,107,193,365,621,1082,1715,2777,4247,6519,9608,

   14210,20263,28957,40411,56307,76929,104907,140431

I agree about polyominoes being simply connected by edge matching of their unit elements. 

My counter-example is valuable as a "caveat emptor" because it shows the con/crete intersection

method has some pathological cases outside the set of what we actually consider valid. 

All the best, 

--Brad