Matlab Error using vertcat Out of memory

Question

I am running the following code. It works for smaller NumberOfVariables, but not for any for 8 or larger due to lack of memory. I really just need the first AllAnswers that does not contain zero, but would like all AllAnswers if possible.

NumberOfVariables = 9;
k=NumberOfVariables^2-NumberOfVariables+1;
integers = 0:k-1;
numbers = 1:k-1;
tic
s = combnk(integers,NumberOfVariables);
AllAnswers = [];
for i = 1:size(s,1)
G=combnk(s(i,:),2);
G = [(G(:,1)'-G(:,2)') (G(:,2)'-G(:,1)')];
G = sort(mod(G,k));
if (isequal(G,numbers))
AllAnswers = [AllAnswers;s(i,:)];
end
end
toc

Answer 1

 s = combnk(integers,NumberOfVariables);

Is the list of all Number of Variables -sized combinations of elements in integers . So that's a list with

k!/((NumberOfVariables!(k-NumberOfVariables)!)

or, since k = NumberOfVariables^2-NumberOfVariables

(NumberOfVariables^2-NumberOfVariables)!/((NumberOfVariables!(NumberOfVariables^2-2*NumberOfVariables)!)

using x for NumberOfVariables, to keep this readable:

      (x²-x)!        x²!            x²!            x²!     
#s= ----------  > ---------- > ------------ = ------------ 
     x!(x²-2x)!   x!(x²-2x)!   x!(x²-2x+1)!    x!((x-1)²!)

you get the idea. this thing is not your friend if you're actually approaching it by allocating memory for #s elements. In fact, looking at this, for x>=4, this will grow faster than e^x.

For NumberOfVariables starting at 10, that simply gets incredibly large.

Do the math!

https://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427euo0stpao37