This is a copy of my post at Mathexchange.com, as my question is still not fully answered and I really wanna find a solution to this. Feel free to refer to there for useful comments and partial solutions:
Number of ways to connect sets of $k$ dots in a perfect $n$-gon
Let Q(n,k) be the number of ways in which we can connect sets of k vertices (dots), in a given perfect n-gon, such that no two lines intersect at the interior of the n-gon and no vertice remains isolated.
Intersection of the lines outisde the n-gon is acceptable. Obviously, k|n, and n can't be prime because otherwise there will be dots left unconnected. The n-gon itself is an acceptable solution to a connection of n vertices, and in the case of k>2, these aren't lines, but a set of connected lines, a sort of a network formed by connected planar graphs with straight edges with k vertices, which are required to be vertices of the n-gon itself.
There must always be S=n/k sets of lines. For k=2, there are exactly n/k lines, and for k>2, there are exactly n/k, not lines but sets of such connected planar graphs.
Take for example Q(6,2). We have a perfect hexagon. By brute-forcing with pencil and paper, I found that there are 5 ways to connect sets of 2 vertices (dots) such that no two lines intersect inside the hexagon. Hence, Q(6,2) = 5.
The following image depicts the case of Q(6,2):
Now let's move one step further:
Let U(n,k) be the number of unique ways to connect sets of k dots/vertices in a perfect n-gon, such that no lines intersect inside the n-gon, and rotational symmetry is neglected, i.e, every possible arrangement is unique and can't be formed by rotating another arrangement in any way. U(6,2)=2. Note that U(6,2)=2 because the arrangements of the first line in the image are not unique, and can be formed by rotating one another. The same happens for the second line of arrangements. Hence U(6,2)=2.
I'm pretty sure there's a pure combinatorial approach to this problem, perhaps involving Polya's Enumeration Theorem (PET). Is there an elegant solution to these functions? Can they even be solved for k>2?
Any light shed on any of the functions will be very much appreciable, as I haven't been successful in deriving a formula for any of them. Both algorithms and formulas will be great!
P.S: I can program in java and mathematica
Thanks a lot in advance.
EDIT - Temporary Solutions + Relevant questions AND progress
Q(n,2) = C(n/2) where C(n) = 1/(n+1) * Binomial (2n , n) And C(n) denotes the n'th Catalan number.
Now let us denote W(n) = U(n,2). Can you find a formula for W(n)? Perhaps a connection between Q(n,2) and W(n)?
