Adjacent Elements in MATLAB with Mathematical Formulation

Question

I have a set with elements and the possible adjacent combinations for this are:

So the total possible combinations are c=11 which can be calculated with the formula:

I can model this using a as below whose elements can be represented as a(n,c) are:

I have tried to implement this in MATLAB, but since I have hard-coded the above math my code is not extensible for cases where n > 4 :

n=4;
c=((n^2)/2)+(n/2)+1;
A=zeros(n,c); 

for i=1:n 
    A(i,i+1)=1; 
end 

for i=1:n-1 
    A(i,n+i+1)=1;
    A(i+1,n+i+1)=1;
end 

for i=1:n-2 
    A(i,n+i+4)=1;
    A(i+1,n+i+4)=1;
    A(i+2,n+i+4)=1; 
end 

for i=1:n-3 
    A(i,n+i+6)=1;
    A(i+1,n+i+6)=1;
    A(i+2,n+i+6)=1;
    A(i+3,n+i+6)=1;
end

Is there a relatively low complexity method to transform this problem in MATLAB with n number of elements of set N , following my above mathematical formulation?

Answer 1

The easy way to go about this is to take a bit pattern with the first k bits set and shift it down n - k times, saving each shifted column vector to the result. So, starting from

1
0
0
0

Shift 1, 2, and 3 times to get

|1 0 0 0|
|0 1 0 0|
|0 0 1 0|
|0 0 0 1|

We'll use circshift to achieve this.

function A = adjcombs(n)
   c = (n^2 + n)/2 + 1;   % number of combinations
   A = zeros(n,c);        % preallocate output array 

   col_idx = 1;             % skip the first (all-zero) column 
   curr_col = zeros(n,1);   % column vector containing current combination
   for elem_count = 1:n
      curr_col(elem_count) = 1;   % add another element to our combination
      for shift_count = 0:(n - elem_count)
         col_idx = col_idx + 1;   % increment column index 
         % shift the current column and insert it at the proper index
         A(:,col_idx) = circshift(curr_col, shift_count);
      end
   end
end

Calling the function with n = 4 and 6 we get:

>> A = adjcombs(4)
A =

   0   1   0   0   0   1   0   0   1   0   1
   0   0   1   0   0   1   1   0   1   1   1
   0   0   0   1   0   0   1   1   1   1   1
   0   0   0   0   1   0   0   1   0   1   1

>> A = adjcombs(6)
A =

   0   1   0   0   0   0   0   1   0   0   0   0   1   0   0   0   1   0   0   1   0   1
   0   0   1   0   0   0   0   1   1   0   0   0   1   1   0   0   1   1   0   1   1   1
   0   0   0   1   0   0   0   0   1   1   0   0   1   1   1   0   1   1   1   1   1   1
   0   0   0   0   1   0   0   0   0   1   1   0   0   1   1   1   1   1   1   1   1   1
   0   0   0   0   0   1   0   0   0   0   1   1   0   0   1   1   0   1   1   1   1   1
   0   0   0   0   0   0   1   0   0   0   0   1   0   0   0   1   0   0   1   0   1   1