- #1
Nick O
- 158
- 8
(I don't like the title, since it is a bit misleading. But, I couldn't think of a more descriptive title that fit in the length restrictions.)
A recurring theme in a problem I am exploring is counting the number of subsets of size n in [itex]Z^{d}_{3}[/itex] that have at least m mutually cohyperplanar (dimension d-1) points.
For example, if n=5, m=3, d=2, the question is: "How many ways are there to select 5 points from a 3x3 plane such that any 3 points are mutually collinear?" Because 5 points guarantee a complete line in [itex]Z^{2}_{3}[/itex], the answer is the same as 9 choose 5, specifically 126.
The specific case I am currently interested in, but most likely not the last, is the case with m=7, d=3. So, the question is: "How many ways are there to select n points from [itex]Z^{3}_{3}[/itex] such that any 7 are mutually coplanar?", where n is a variable.
Is it too much to expect that there should be a fairly simple combinatorial answer to this? So far I have not been able to derive it, but I feel that it should exist.
A recurring theme in a problem I am exploring is counting the number of subsets of size n in [itex]Z^{d}_{3}[/itex] that have at least m mutually cohyperplanar (dimension d-1) points.
For example, if n=5, m=3, d=2, the question is: "How many ways are there to select 5 points from a 3x3 plane such that any 3 points are mutually collinear?" Because 5 points guarantee a complete line in [itex]Z^{2}_{3}[/itex], the answer is the same as 9 choose 5, specifically 126.
The specific case I am currently interested in, but most likely not the last, is the case with m=7, d=3. So, the question is: "How many ways are there to select n points from [itex]Z^{3}_{3}[/itex] such that any 7 are mutually coplanar?", where n is a variable.
Is it too much to expect that there should be a fairly simple combinatorial answer to this? So far I have not been able to derive it, but I feel that it should exist.