[seqfan] Re: L-connected polyominoes

[John Mason]

Allan
Would the following be a definition of what you are looking for?
"a(n) is the number of free polyominoes such that there is an L-shaped path
between any pair of cells, consisting of a horizontal arm of x >= 0 cells
contained within the polyomino, and a vertical leg of y >= 0 cells
similarly contained."

If so, I propose the following figures for n through to 18:

1, 1, 2, 4, 7, 14, 24, 48, 83, 155, 265, 472, 793, 1356, 2235, 3700, 5977,
9636

They are a subset of the convex polyominoes (A359661).

Below are the 14 hexominoes (best seen with Courier New) that I count. Tell
me if your count disagrees.

john

O__
OO_
OOO


OO__
OOOO


O___
O___
OOOO


OOO
OOO


O__
OOO
OO_


_OO_
OOOO


_O__
_O__
OOOO


_O_
OOO
OO_


O____
OOOOO


_O___
OOOOO


__O__
OOOOO


O___
OOOO
O___


_O__
OOOO
_O__


OOOOOO





On Mon, Jan 23, 2023 at 6:49 AM Allan Wechsler <acwacw at gmail.com> wrote:

> The following seems like as idea that must be in OEIS already, but I have
> been unable to assemble enough data (that I actually believe) to find it.
>
> In some free polyominoes, every pair of cells is part of an "L" that is
> also part of the polyomino. The smallest polyomino that *isn't *connected
> in this way is the skew tetromino, whose end cells cannot be connected by
> an "L".
>
> I am pretty sure that the census of L-connected polyominoes begins: 1
> monomino; 1 domino; 2 trominoes; 4 tetrominoes; and 7 pentominoes (out of
> 12). I am not sure about hexominoes, but my current best guess is 13. OEIS
> reports dozens of matches to this sequence of values.
>
> Do any of the assembled intellects of SeqFan remember running into
> something like this? Or is your searching ability better than mine? (Or can
> you compute the number of hexominoes and heptominoes confidently enough to
> nail it down?)
>
> Thanks in advance!
>
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