TITLE: What does it mean when a Sigma has two variables at the bottom
QUESTION [5 upvotes]: In a paper I'm reading there is a sigma notation that I'm not understanding. They have two variables at the bottom of the sigma.
From the paper:
A finite set of requirements Req = {r1,...,rn} and D is a distribution,
satisfying the following normalization property:
$$
\sum\limits_{r_i,r_j} D(r_i,r_j) = 1
$$
So does that mean that we are going to sum all of the combinations of ri and rj where i != j?  So if n=5 we would have (5 choose 2) = 10 combinations to sum.  Is that right?
Thank you.

REPLY [9 votes]: You have $$\sum\limits_{r_i,r_j} D(r_i,r_j) = \sum_{i=1}^n \sum_{j=1}^n D(r_i,r_j)$$
Which means that the sum is over all combinations $(i,j) \in \{1, \dots ,n\}^2$. Not only the ones for which $i \neq j$. The notation for this last case would be $$\sum_{i \neq j} D(r_i,r_j).$$