Discussion
Tue Jul 23
02:56
Michel Marcus: number of numbers ?
04:18
Ya-Ping Lu: Yes, it's the number of "numbers of triangles". For example, for n = 3, the possible numbers of triangles with n+2=5 vertices are 2, 3, 4, or 5. So, the number of numbers of triangles for n = 3 is 4 and a(3) = 4. Note that, for the same number of triangles, there could be more than one 1 graph.
11:13
Andrew Howroyd: How do you get to 2 triangles with 5 vertices? I presume by "planar connected graphs of triangles" you mean a planar triangulation - but this has 6 faces (triangles) when there are 5 vertices - I can't see how to get rid of 4 of them to make 2. As it stands this comment will not be easily understood.
11:18
Andrew Howroyd: Perhaps by "triangle" you mean face (with any number of sides)?
11:24
Andrew Howroyd: Is there a table (tabf/tabl) that this is the number of non-zero terms in a row/column? (something like A049334, but for whatever problem you are talking about).
22:54
Ya-Ping Lu: With 5 vertices, you can do it by sharing 1 vertex between 2 triangles.
"planar triangulation" may be a better description than "planar connected graphs of triangles".
Here is a list of # of vertices, and possible #s of triangles for n up to 7:
n # of vertices #s of triangles a(n)
-- -------------- ---------------------------- ------
1 3 1 1
2 4 2,3 2
3 5 2,3,4,5 4
4 6 3,4,5,6,7 5
5 7 3,4,5,6,7,8,9 7
6 8 4,5,6,7,8,9,10,11 8
7 9 4,5,6,7,8,9,10,11,12,13 10
Wed Jul 24
08:53
Andrew Howroyd: Let's go back to the beginning. What is the sequence that counts "planar connected graphs of triangles" with n vertices? You cannot just invent terminology like this or noone will know what you mean (especially as what you are describing is probably not even a graph but rather an embedding)
Thu Jul 25
05:48
Ya-Ping Lu: By "planar connected graphs of triangles", I mean planar connected graphs containing triangles only. In other words, any vertex or edge should belong to a triangle. I know this type of graph is not "triangulate graph", but not sure what the right terminology should be. Any suggestion? Thanks!
11:17
Andrew Howroyd: This is utter nonsense. You don't seem to have a clear understanding of the difference between a graph and a drawing of a graph. You also don't seem to know what a triangle is in graph theory. The only graphs containing 'triangles only' are the complete graphs. I have asked you for the sequence that counts "planar connected graphs of triangles" with n vertices" so that we can we can find the correct terminology. Instead of answering my question you have evaded. I suggest you start by reading a book on graph theory and also studying sequences in oeis that talk about graphs and maps so that you get a feel for the correct terminology. Otherwise what you are during is basically vandalism, like writing graffiti. In any case what you are proposing here is clearly wrong. With 4 vertices, the complete graph is planar and connected - this graph has 4 triangles, yet you indicate the maximum number of triangles in 3.
11:18
Andrew Howroyd: Until you answer my 08:53 question with a sequence number, I really can't help you.
Sun Jul 28
04:57
Ya-Ping Lu: I'm sorry about the confusion caused by my comments. I'm trying my best to communicate with the right terminology. It would be much easier if I could provide you some drawings of the graph I am talking about. I think the first 4 terms of the sequence that counts the number of graphs that I described with n vertices are: a(3) = 1, a(4) = 2, a(5) = 5, a(6) = 19. The possible sequence could be A328977, but I am not sure.
Does if make sense if the description is changed to something like "a(n) is the number of numbers of inner faces of planar connected graphs with n+2 nodes in which all the inner faces have 3 edges"? Thanks!
Mon Jul 29
05:29
Ya-Ping Lu: Here are the degree sequences (DS), numbers of vertices (V), numbers of edges (E), and numbers of inner faces (F) of the graphs in the sequence mentioned above. The sequence in parentheses is the list of the numbers of inner faces associated with the vertices in the graph, to distinguish the graphs with the same degree sequence from each other.
n a(n) DS V E F
-- ----- ------------- ------- ------- -------
3 1 222 3 3 1
4 2 2233 4 5 2
3333 4 6 3
5 5 22224 5 6 2
22334 5 7 3
23344 5 8 4
33334 5 8 4
33444 5 9 5
6 19 222235 6 8 3
222334 6 8 3
222444 6 9 4
223335 6 9 4 (112224)
223335 6 9 4 (112233)
223344 6 9 4
223445 6 10 5
233345 6 10 5
233444 6 10 5
333335 6 10 5
233455 6 11 6
234445 6 11 6
333355 6 11 6
333445 6 11 6
334444 6 11 6
334455 6 12 7 (233445)
334455 6 12 7 (234444)
334455 6 12 7 (333345)
444444 6 12 7
11:53
Andrew Howroyd: Firstly, 1,2,5,19,.... is certainly not A328977. A328977 only includes 2-connected graphs. Your first example for a(5) with degree sequence 22224 is not 2-connected.
Regardless, this is not the correct sequence to be adding this information to oeis and by trying to put it here you are making a mess. I will try to explain since this is an important point that you need to understand. There are many sequences that are irregular triangles (search keyword:tabf). A great many of these triangles have widths that are 2*n, floor(n/2), floor(3*n/2) and other simple proportions. Others also have a sub-triangle of zeros. Because there are thousands of such triangles it doesn't make sense to include information about the minimum, maximum and number of nonzero terms of each triangle in simple generic sequences like this (A001651, floor(n/2), etc), because the comments would be huge for one thing but also because it would be nearly impossible to distinguish one comment from another. The natural place for these types of comments is in the triangles themselves. I hope that makes sense.
Your edit to A080513 is even worse because that sequence could just as well be floor(n/2), ceiling(n/2) - there are several sequences with almost the exact same terms. This is an an extremely generic sequence. There are thousands of triangles that have a width of ceiling(n/2).
If you are interested in this topic, then what you first need to do is continue the development of 1,2,5,19 and create a sequence for that if it does not already exist in oeis. [You might be able to use the program plantri to calculate these terms - search plantri in oeis to find many sequences]. (Although plantri can generate triangulations of a polygon, I don't know if it extends this to 1-connected which is what you have here - perhaps that explains why not already in oeis. You would need to investigate; and even if plantri doesn't give you exactly what you want you might be able to use it to create a starting point). Once you have that sequence and have added it to oeis you can then do a breakdown by numbers of internal triangular faces (a tabf sequence). That will be the correct sequence to include information about minimum and maximum number of triangles. (instead of scattering amongst sequences like this). Here it is no good to anyone - it is just making a mess.
Sun Aug 11
14:48
N. J. A. Sloane: This is unsatisfactory, and I will reject this comment.
