If we have a matrix with N elements, where each element can take values G ( 0-255), we can obtain 256power N possibilities of matrixes.
The derivative of each matrix is calculated as follows:

S=∑_(n=1)^(N-1)▒df(n)/dx=∑_(n=1)^(N-1)〖| f(n+1)-f(n)|〗

Since 0≤ df(n)/dx ≤255 the minimum and maximum values of s are:

max S=(N-1)×255
min S= 0

I need to find how many matrixes have the same S.

Can anybody help me?
Thank you.