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].

Answer 2

Having a number
x = 2^k₁+2^k₂+...+2^k_m
where k₁<k₂<...<k_m
it could be claimed that position of number x in lexicographically ordered sequence of all numbers with the same hamming weight is
lex_order(x) = C(k₁,1)+C(k₂,2)+...+C(k_m,m)
where C(n,m) = n!/m!/(n-m)! or 0 if m>n

Example:

3 = 2⁰ + 2¹
lex_order(3) = C(0,1)+C(1,2) = 0+0 = 0

5 = 2⁰ + 2²
lex_order(5) = C(0,1)+C(2,2) = 0+1 = 1

6 = 2¹ + 2²
lex_order(6) = C(1,1)+C(2,2) = 1+1 = 2

9 = 2⁰ + 2³
lex_order(9) = C(0,1)+C(3,2) = 0+3 = 3