[seqfan] Re: Counting polyominoes with a given "sprawl"

Frank Adams-watters franktaw at netscape.net
Mon Oct 12 00:40:20 CEST 2020


Another generalization is to iterate counting the number of cells adjacent to any preceding step. This gives a sequence for each starting polyomino. (I wouldn't think we would more than one of these, probably the R pentomino.)

Rather than the "sprawl", I think "blur" might be a better word for this.

Franklin T. Adams-Watters


-----Original Message-----
From: Sean A. Irvine <sairvin at gmail.com>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Sent: Sun, Oct 11, 2020 2:36 pm
Subject: [seqfan] Re: Counting polyominoes with a given "sprawl"

Hi Allan,

I haven't looked too closely at exactly what you write, but there are a
number of existing sequences which attempt to capture this general idea in
a variety of ways.

One way is to compute the "perimeter" of the polyomino.  I think the
perimeter is very close to your sprawl, but does not include the cells of
the polyomino itself.  This definitely has a probabilistic interpretation
in physics.  An example of this is in A003203.

There are also various sequence measuring diameter etc.

Sean.



On Mon, 12 Oct 2020 at 08:26, Allan Wechsler <acwacw at gmail.com> wrote:

> The classic A000105 counts the number of polyominoes with a given number of
> cells.
>
> Define the "sprawl" of a polyomino to be the number of cells either in the
> polyomino or edge-adjacent to it, when the polyomino is drawn on a piece of
> graph paper.
>
> For example, the R-pentomino has five cells, and is adjacent to nine more,
> so it has a sprawl of 14.
>
> The sprawl is important when you are trying to calculate how likely it is
> to find a given polyomino on a field of black and white cells, randomly
> colored with equal probability. (There are other terms, but the sprawl is
> important.)
>
> How many polyominoes are there with a sprawl n? Starting with n = 3, I am
> pretty sure that this sequence starts 0,0,1,0,0,1,0,1,1,3,2, and if this
> data is right, then the sequence is not yet archived in OEIS.
>
> It's a very obvious idea, and so I will be less surprised if I simply
> counted wrong. Can anyone confirm these numbers?
>
> For n = 0, 1, or 2 there are definitional problems which permit argument
> about how the sequence gets going, but from n = 3 onward things seem fairly
> clear.
>
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