4 ms·

Polyominoes have been obsessing me. Not tiling but counting them. No formula has been yet found. Their free numbers (i.e up to rotations and flipping) go 1,

Polyominoes have been obsessing me. Not tiling but counting them. No formula has been yet found. Their free numbers (i.e up to rotations and flipping) go 1, 1, 2, 5, 12, 35, 108, 369, 1285, ... Meaning for ex. there are 5 tetrominoes : ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ The myterious sequence follows a ~4 growth rate. I've been (nowhere as a mathematician) exploring this problem for years, generating them and trying in vain to find patterns in their properties. I'd love readers with a high view in combinatorics telling what they think of this problem. Do you think a formula will one day be found or rather that there can't be a closed one for some necessary reason ?

lkozma 4y ago
This is a very nice and deep question. Are you aware of the published research on the problem? The growth rate is indeed around 4 and now known to be strictly above 4: https://page.mi.fu-berlin.de/rote/Papers/pdf/Lambda-4.pdf https://page.mi.fu-berlin.de/rote/Papers/pdf/Lambda-4.pdf If I remember correctly, there was a non-rigorous argument for a concrete conjectured value not far above 4.
- alcover 4y ago
  Thanks for the link. we obtain 4.00253176 as a certified lower bound How intringing it's strictly just above 4 ! Maybe it has to do with symmetries. The proportion of symmetric polys in successive generations goes towards zero. I suspect that this 'pollutes' the asymptot. About published research, I just glanced at the little I found, (as I often can't do more than glance, being limited in Maths.) Also I prefer to go at it naively first.
slazaro 4y ago
First thing that came to mind is, maybe if you don't eliminate symmetries or roations or both, that maybe a clear formula appears? Maybe it's one of those steps that makes it a hard problem.
- alcover 4y ago
  Alas, a formula for what you suggest (called fixed polyominoes) seems also out of reach.

lkozma 4y ago

This is a very nice and deep question. Are you aware of the published research on the problem? The growth rate is indeed around 4 and now known to be strictly above 4: https://page.mi.fu-berlin.de/rote/Papers/pdf/Lambda-4.pdf https://page.mi.fu-berlin.de/rote/Papers/pdf/Lambda-4.pdf If I remember correctly, there was a non-rigorous argument for a concrete conjectured value not far above 4.

alcover 4y ago
Thanks for the link. we obtain 4.00253176 as a certified lower bound How intringing it's strictly just above 4 ! Maybe it has to do with symmetries. The proportion of symmetric polys in successive generations goes towards zero. I suspect that this 'pollutes' the asymptot. About published research, I just glanced at the little I found, (as I often can't do more than glance, being limited in Maths.) Also I prefer to go at it naively first.

slazaro 4y ago

First thing that came to mind is, maybe if you don't eliminate symmetries or roations or both, that maybe a clear formula appears? Maybe it's one of those steps that makes it a hard problem.

alcover 4y ago
Alas, a formula for what you suggest (called fixed polyominoes) seems also out of reach.