[seqfan] Re: L-connected polyominoes

[Allan Wechsler]

Tom Duff: your confusions are completely warranted, though John Mason
seemed to be able to read my mind. An "L-shaped path" through a polyomino
is a chain of cells, all contained in the polyomino, that consists of a
medial cell and two straight chains leading off from it in two orthogonal
directions. These side chains may be of any lengths A and B, and then the
total number of cells in the path is A + B + 1. The "skew tetromino" is the
tetromino obtained from a 2x3 hexomino by removing two diagonally opposite
corner cells. It is sometimes called "S" or "Z". In a binary-based code we
sometimes use for describing Life patterns, the skew tetromino is <132>.

John Mason: despite my lack of clarity, you were able to read my mind
correctly. My hand counts agree with your numbers through n = 7, which
gives me a lot of confidence. And Superseeker came up empty on it, so I
suppose I will submit. Do you want co-authorial credit? Is your code
concise enough to include in the entry?

Others: if anybody else can confirm John Mason's data, that would be lovely.

On Mon, Jan 23, 2023 at 5:22 AM John Mason <masonmilan33 at gmail.com> wrote:

> 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!
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>