[seqfan] Re: Large Chambers in Lattice Polygons

Neil Sloane njasloane at gmail.com
Wed Jun 7 01:40:49 CEST 2017


Hugo, bravo!

Please submit them all!

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com


On Tue, Jun 6, 2017 at 6:22 PM, Hugo Pfoertner <yae9911 at gmail.com> wrote:

> Dear SeqFans,
>
> when searching for information on lattice polygons, I've found the
> following web page created in 2001 and 2004 by Marc E. Pfetsch and Guenter
> M. Ziegler.
>
> http://www.mathematik.tu-darmstadt.de/~pfetsch/chambers/
>
> Besides from the interesting pictures and the fascinating connection to the
> article on Supernormal vector Configurations by Serkan Hosten
> <https://arxiv.org/find/math/1/au:+Hosten_S/0/1/0/all/0/1>, Diane Maclagan
> <https://arxiv.org/find/math/1/au:+Maclagan_D/0/1/0/all/0/1>, Bernd
> Sturmfels <https://arxiv.org/find/math/1/au:+Sturmfels_B/0/1/0/all/0/1>
> https://arxiv.org/abs/math/0105036
>
> the page has several tables which might be worthwhile to become new
> sequences. I have therefore grabbed 6 new A-numbers and have started to
> prepare the following drafts:
>
> Some illustrations might be useful:
> http://www.randomwalk.de/sequences/chambers.pdf
> http://www.randomwalk.de/sequences/cc_intersect.pdf
>
>
> https://oeis.org/draft/A288177
> NAME
>
> Maximum number of vertices of any convex polygon formed by drawing all line
> segments connecting any two lattice points of an n X m convex lattice
> polygon in the plane.
> DATA
>
> 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 4, 5, 5, 6, 6, 4, 5, 5, 6, 6, 6, 4, 5, 6, 6,
> 6, 7, 7, 4, 5, 7, 6, 7, 7, 7, 7, 4, 5, 6, 6, 7, 7, 8, 8, 8, 4, 5, 6, 6, 7,
> 7, 8, 8, 8, 7, 4, 5, 6, 6, 7, 7, 8, 8, 8, 8, 8, 4, 5, 7, 6, 7, 7, 8, 7, 8,
> 8, 8, 8, 4, 5, 8, 6, 7, 7, 8, 7, 8, 8, 8, 8, 8, 4, 5, 8, 6, 7, 7, 8, 8, 8,
> 8, 8, 8, 8, 8, 4, 5, 8, 6, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 4, 5, 7, 6, 7,
> 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 4, 5, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9,
> 9, 9, 9, 4, 5, 8, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 9
> OFFSET
>
> 1,1
> COMMENTS
>
> The table is given in the section "Results" of the notes by M. E. Pfetsch
> and G. M. Ziegler, see link.
> LINKS
>
> Serkan Hosten, Diane Maclagan, Bernd Sturmfels, <a href="
> https://arxiv.org/abs/math/0105036">Supernormal Vector Configurations</a>,
> arXiv:math/0105036 [math.CO], 4 May 2001
>
> Marc E. Pfetsch, Günter M. Ziegler, <a href="
> http://www.mathematik.tu-darmstadt.de/~pfetsch/chambers/">Large Chambers
> in
> a Lattice Polygon</a> (Notes), March 28, 2001, December 13, 2004
>
> Hugo Pfoertner, <a href="/A288177
> <https://oeis.org/A288177>/a288177.pdf">Illustrations
> of Chamber Complexes up to 5 X 5</a>.
> EXAMPLE
>
> Drawing the diagonals in a lattice square of size 1X1 produces 4 triangles,
> so a(1)=3.
> CROSSREFS
>
> Cf. A288178 <https://oeis.org/A288178> (diagonal of table), A288179
> <https://oeis.org/A288179>, A288180 <https://oeis.org/A288180>, A288181
> <https://oeis.org/A288181>
> KEYWORD
> nonn,tabl,changed
>
>
> https://oeis.org/draft/A288178
> NAME
>
> Sizes of largest chambers in an n X n lattice complex. Diagonal of table
> given in A288177 <https://oeis.org/A288177>.
> DATA
>
> 3, 4, 4, 5, 6, 6, 7, 7, 8, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10,
> 10, 10
> OFFSET
>
> 1,1
> COMMENTS
>
> For comments, references and links see A288177 <https://oeis.org/A288177>.
> In addition to the table given there, data for n>18 are provided here.
> CROSSREFS
>
> Cf. A288177 <https://oeis.org/A288177>.
> KEYWORD
>
> nonn,changed
>
> https://oeis.org/draft/A288179
>
> Maximum number of vertices of any convex polygon formed in the middle
> square
> of the boundary by drawing the line segments connecting any two lattice
> points in an (2k+1) X (2k+1) lattice polygon.
> DATA
>
> 4, 6, 7, 6, 7, 6, 6, 7, 7, 8, 8, 7, 7, 8, 8, 8, 8, 8, 8
> OFFSET
>
> 1,1
> COMMENTS
>
> For comments, references and links see A288177 <https://oeis.org/A288177>.
> CROSSREFS
>
> Cf. A288177 <https://oeis.org/A288177>.
> KEYWORD
>
> nonn,changed
>
>
> Those first 3 sequences are just material copied from the mentioned web
> page. Since I'm unsure about the nomenclature, I'd like to get some
> feedback before submitting them.
>
> The next 3 sequences are proposals for some additional counting that should
> be done for the chamber complex. Most of the numbers were created by hand,
> and I'd appreciate if someone could do some checking of the numbers,
> ideally by a program.
>
> https://oeis.org/draft/A288180
> NAME
>
> Number of intersection points formed by drawing the line segments
> connecting
> any two lattice points of an n X m convex lattice polygon.
> DATA
>
> 5, 13, 37, 35, 123, 355
> OFFSET
>
> 1,1
> COMMENTS
>
> If more than two lines intersect in the same point, only one intersection
> is counted.
>
> Conjectured next table row for n=4,m=1...4: 75, 269, 775,?1764?
>
> n=5,m=1,2,3: 159, 592, ?1765?
> REFERENCES
>
> For references and links see A288177 <https://oeis.org/A288177>.
> LINKS
>
> Hugo Pfoertner, <a href="/A288180
> <https://oeis.org/A288180>/a288180.pdf">Illustration
> of intersection points up to 5 X 3</a>.
> CROSSREFS
>
> Cf. A288177 <https://oeis.org/A288177>.
> KEYWORD
>
> nonn,tabl,changed
>
> https://oeis.org/draft/A288181
>
> This would definitely need a program for counting. The web page provides
> two low quality example pictures, one for 5X5 and the other one for 7X5,
> from which it is impossible to retrieve the chamber count.
> Even with better resolution manual counting would be extremely cumbersome:
> http://www.randomwalk.de/sequences/A288181_75.pdf
>
> NAME
>
> Occurrence counts of chambers with maximum number of vertices in the
> chamber
> complex of an n X m lattice polygon as described in A288177
> <https://oeis.org/A288177>.
> DATA
>
> 4, 2, 8, 14, 54, 168
> OFFSET
>
> 1,1
> LINKS
>
> Hugo Pfoertner, <a href="/A288181
> <https://oeis.org/A288181>/a288181.pdf">Chamber
> complex of 5 X 5 lattice polygon.</a> Illustration.
> EXAMPLE
>
> The chamber complex of the 5 X 5 lattice polygon has 16 chambers of size 6,
> so a(15)=16.
> CROSSREFS
>
> Cf. 288177.
> KEYWORD
>
> nonn,tabl,changed
>
>
> The total number of all chambers also would be an interesting programming
> exercise, as one can see from this 9X5 example picture:
>
> http://www.randomwalk.de/sequences/A288187_95.pdf
>
>
> https://oeis.org/draft/A288187
>
>
> NAME
>
> Number of polygons formed by drawing the line segments connecting any two
> of
> the (n+1)*(m+1) lattice points in an n * m lattice polygon.
> DATA
>
> 4, 16, 56, 46, 176, 516
> OFFSET
>
> 1,1
> COMMENTS
>
> Polygons are counted irrespective of their numbers of vertices.
> REFERENCES
>
> For references and links see A288177 <https://oeis.org/A288177>.
> EXAMPLE
>
> The diagonals of the 1 X 1 lattice polygon, i.e. the square, cuts it into 4
> triangles. Therefore a(1)=4.
> CROSSREFS
>
> Cf. A288177 <https://oeis.org/A288177>, A288180 <https://oeis.org/A288180
> >,
> A288181 <https://oeis.org/A288181>.
> KEYWORD
>
> nonn,tabl,changed
>
>
>
> Any feedback is welcome.
>
> Hugo Pfoertner
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>



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