[seqfan] A000081 & nonoverlapping circles(?)

[Jim Nastos]

On Wed, 11 Jun 2003, r.shepherd wrote:

> Can anyone confirm that this comment in A81 below is correct?
> Trying to see it for myself I've created a .jpg file that shows
> 25 arrangements of 5 nonoverlapping circles -- not A000081(6)=20.

  I haven't drawn out the configurations, but I think I can convince you 
that 20 is correct:
  A(n) = number of such configurations with n circles. The sequence says 
A(5) = 20 and your claim is A(5) = 25.

  Any configuration of 5 circles that has an isolated circle can be 
represented as the isolated circle with any of the A(4)=9 configurations 
for the remainder. So we have contributed 9 to the total.
  Similarly, any configuration of 5 circles that has a huge circle 
engulfing all the rest can be represented as the one big circle with all 
A(4)=9 configurations on its interior. This contributed another 9 to our 
total.
  The remaining case has no isolated circle and no all-engulfing circle, 
so we have two or more circles with non-empty interiors. Note that it is 
impossible to have 3 nonempty circles, so we must have exactly 2; one of 
which has 2 interior circles and the other has 1. The number of 
configurations of the two interiors are 2 and 1, so this contributed 2x1=2 
to the total.
  9 + 9 + 2 = 20, and every configuration of 5 circles must fall under one 
of these three categories.

Jim