What is "axially"?

[Joshua Zucker]

On Fri, Mar 7, 2008 at 2:43 PM, Tanya Khovanova
<mathoflove-seqfan at yahoo.com> wrote:
> Dear SeqFans,
>
>  I am confused by the word "axially". What does it mean in:
>  A006746                  Axially symmetric polyominoes with n cells.
>
>  As opposed to:
>  A030227                  Number of n-celled polyominoes with bilateral symmetry.
>
>


Axially should mean that it has an axis of (rotation) symmetry, as
opposed to bilateral (two-sided) reflection symmetry.

It's not clear whether "axially" means symmetric under a 90 degree
rotation or 180; I'm sure that'll be clear enough from the context,
though.

let's see, with 4 the axially symmetric ones should be
xx
xx

xxxx

 xx
xx  (if it's 180, but not 90)


But A006746 says a(4) = 1 so maybe it's talking only about
xx
xx

But then a(5) = 2 ... so which 2 are they?

OK, now I'm confused too.  Help!

Hm, wait a minute.  a(6) = 6, and according to
  http://en.wikipedia.org/wiki/Image:Hexominoes.svg
there are 6 hexominoes with ONE line of reflection symmetry oriented
with the gridlines.

So maybe that's what they mean after all, in which case "axial" seems
like a bad choice to describe the symmetry.

Then according to A030227, A030227(6) = 10, which means it looks like
it's talking about all the hexominoes that have AT LEAST one line of
reflection symmetry, not necessarily aligned with the gridlines.

Checking n = 7 seems to confirm this.

I think it would be a lot clearer to have A006746 be retitled as
something like "polyominoes with one axis of reflection symmetry,
parallel to the gridlines".


--Joshua