[seqfan] Re: Counting polyominoes with a given "sprawl"
acwacw at gmail.com
Mon Oct 12 00:42:36 CEST 2020
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.
> 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
> > cells.
> > Define the "sprawl" of a polyomino to be the number of cells either in
> > polyomino or edge-adjacent to it, when the polyomino is drawn on a piece
> > graph paper.
> > For example, the R-pentomino has five cells, and is adjacent to nine
> > 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
> > clear.
> > --
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