Determin the lexicographic distance between two integers

Question

Say we have the lexicographicaly integers 3,5,6,9,10,12 or 0011,0101,0110,1001,1010,1100 Each with two bits set.

What I want is to find the distance(how

Answer 1

If a and b are the positions of the two set bits, with zero being the least significant position, and a always being greater than b, then you can calculate:

n = a*(a-1)/2 + b

and the distance between two values is the difference between the two n values.

Example:

3->12:
  3:  a1=1, b1=0, n1=0
  12: a2=3, b2=2, n2=5
  answer: n2-n1 = 5

To extend this to other hamming weights, you can use this formula:

n = sum{i=1..m}(factorial(position[i])/(factorial(i)*factorial(position[i]-i)))

where m is the hamming weight, and position[i] is the position of the i'th set bit, counting from the least significant bit, with the least significant set bit's position being position[1].