[seqfan] Re: partitions of a circle
Jon Wild
wild at music.mcgill.ca
Thu May 10 19:52:46 CEST 2012
Dear Neil,
This sequence seemed familiar to me and I just found and scanned a closely
related sequence I drew some time ago in an old notebook (I never
submitted it to the oeis).
My rules were: n straight, non-intersecting line segments are drawn
successively in a circle, such that both end points of each segment lie
either on the perimeter of the circle or on a previously drawn line. No
endpoints may coincide. (Thus no "V"s.) My sequence is reduced for mirror
symmetry. For 0 to 4 added lines, I counted 1,1,2,9,63 distinct
configurations. (Under these rules n lines always result in n+1 pieces of
circle, of course.)
http://music.mcgill.ca/~wild/circlePieces.pdf
In the scan linked here, the 9 configurations for n=3 appear down the
left-hand margin. For each 3-config, I found all the 4-configs you can
produce by adding one line segment, except those that had already been
produced from an earlier 3-config. I was quite methodical so I'm
reasonably confident in these results.
Jon Wild
On Mon, 7 May 2012, Neil Sloane wrote:
> Dear Sequence Fans, There is a paper:
> *Valette, G.*; *Zamfirescu, T.* Les partages d'un polygone convexe en
> 4 polygones
> semblables au premier. (French) *J. Combinatorial Theory Ser. B* *16 *(1974),
> 1--16. MR0331217 <http://www.ams.org/mathscinet-getitem?mr=0331217> *(48
> #9551),*
> *see **http://www.sciencedirect.com/science/journal/00958956/16/1,*
> *which studies the ways to divide a polygon into 4 congruent pieces.
> *
>
> But they begin by looking at a simpler question:
> the number of ways to divide a circle into 4 pieces: there are 15 ways,
> according to their rules. Similarly, I think there are 4 ways to divide a
> circle into 3 pieces.
> So there is a sequence 1, 1, 4, 15, ... I can't tell yet if it is in the
> OEIS. I scanned in their
> illustration of the 15 partitions into 4 pieces, and I will be happy to
> send it to anyone
> who wants to try to help find the next couple of terms.
>
> The 4 ways to cut a circle into 3 pieces are:
> 1. draw 2 parallel chords in a circle
> 2. draw a T in a circle
> 3. draw a Y in a circle
> 4. draw a V in a circle
>
> Neil
> --
> Dear Friends, I will soon be retiring from AT&T. New coordinates:
>
> Neil J. A. Sloane, President, OEIS Foundation
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
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