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

Jack Grahl jack.grahl at gmail.com
Tue Oct 13 16:11:04 CEST 2020

```If I've understood correctly, then sprawl is not necessarily the same as
perimeter - number of cells:

The L triomino has two boundary edges adjacent to the same square. Thus the
perimeter is 8 (counting both those edges) but the sprawl is 10 (counting
that square only once).

On Tue, 13 Oct 2020, 15:04 Allan Wechsler, <acwacw at gmail.com> wrote:

> Sean, your interpretation that sprawl = cells + perimeter is correct.
>
> I cannot find a sequence "number of polyominoes with perimeter n" either.
> Starting with n = 3 (the first case that isn't vulnerable to definitional
> quibbling), the sequence should go 0,1,0,1,1,5 ... OEIS has 15 hits for
> this, but none of them have any chance of being the desired sequence. At
> the moment I don't have a good guess for n = 9.
>
> On Sun, Oct 11, 2020 at 3:36 PM Sean A. Irvine <sairvin at gmail.com> wrote:
>
> > 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.
> > >
> > > --
> > > Seqfan Mailing list - http://list.seqfan.eu/
> > >
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>
```