[seqfan] Large Chambers in Lattice Polygons

Wed Jun 7 00:22:45 CEST 2017

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