Adjacent Elements in MATLAB with Mathematical Formulation - algorithm

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