Valid ordered pairs

Given a number N. How to calculate number of pairs (x,y) below N, Such that [S(x)<S(y)] where S(x) and S(y) are sum of digits of x and y respectively?

Example: for N=3, pairs formed such that S(x)<S(y) are:
valid pairs =(0,1) (0,2) (0,3) (1,3) (1,2) (2,3)
invalid pairs=(2,2)(1,1)(3,3) (2,1) (3,1) (0,0) (1,0)

valid pairs =6

Another example N=10

valid pairs= (0,1) (0,2) (0,3) (0,4) (0,5) (0,6) (0,7) (0,8) (0,9)
(1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (1,8) (1,9) (2,3) (2,4) (2,5) (2,6) (2,7) (2,8) (2,9) (3,4) (3,5) (3,6) (3,7) (3,8) (3,9) (4,5) (4,6) (4,7) (4,8) (4,9)
(5,6) (5,7) (5,8) (5,9) (6,7) (6,8) (6,9) (7,8) (7,9) (8,9)     

pairs=45
(1,10) is invalid pair because S(x)=1 and S(y)=1 as [y=10 and S(y)=1+0]