How Many Ways To Subdivide m-gon Using n Cuts?

[Leroy Quet]

I posted this to both sci.math and rec.puzzles, but as of the last I 
checked, there were no replies at all. So I thought that it MIGHT be 
worthy of seq.fan:

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If we have a regular m-gon, how many ways can we pick n vertexes so
that n straight lines from these vertexes, each drawn to a single
interior point P, divides the m-gon into n equal-area pieces?

For example, for m = 6 and n = 3:

(This enumeration might be a little controversial.)

Let the hexagon's vertexes be labeled, in order, from 1 to 6.

3 cases:

1) If we let the vertexes be 1, 3, 5, and P be the hexagon's center,
this works.
Rotating by one vertex gets the same solution rotated.
(2 sets of n vertexes)

2) If we let vertexes be 1, 2, 4, and P be the midpoint of the
edge(4,5), we get a division which corresponds to my puzzle I posted a
little while ago:
http://mathforum.org/discuss/sci.math/a/t/501734
With rotations and reflections we get 12 vertex-sets.
This depends, of course, on exactly what we refer to as an "interior"
point.
If we literally mean "a non-exterior point", then this should be an
acceptable set of vertexes to include in our count.

3) The only other (after rotations and reflections) set of vertexes is
if we have vertexes 1, 2, 3.
Now, here we have an interesting situation. The only place to put P
that I can think of is JUST INSIDE the vertex directly opposite the
center vertex of this set. Placing point P ON the opposing vertex gets
the hexagon divided into 3 equal pieces, BUT one piece is not
simply-connected. And placing P anywhere just inside the opposing
vertex, as to connect the 3rd piece with itself, gets 3 pieces which
are not all equal to 1/3 the hexagon in area.
So, I would count this as not adding anything to the count of total
vertex-sets which divide the hexagon. But some might still. If they
do, 6 vertex-sets are to be added to our total.

So, we have at least 2 vertex-sets, more likely we have 14
vertex-sets, and perhaps 20 vertex-sets, all depending on how we
define the problem


Thanks,
Leroy Quet