Sorry for asking a very dumb question, but in Concrete Mathematics(Graham,Knuth,Patashnik), chapter 2 section 4, Knuth talks about this formula called "Rocky Road".
This is the formula to use when you want to interchange the order of summation of a double sum whose inner sum's range depends on the index variable of the outer sum, like this:
\begin{equation} \displaystyle\sum\limits_{j = 1}^{n}{\displaystyle\sum\limits_{k = j}^{n}{a_j,_k}} \end{equation}
The rocky road formula is as follows: \begin{equation} \displaystyle\sum\limits_{j \in J}^{}{\displaystyle\sum\limits_{k \in K(j)}} = \displaystyle\sum\limits_{k \in K'}{\displaystyle\sum\limits_{j \in J'(k)}} \end{equation}
With the requirement that the sets $J,K(j),K', \text{and} J'(k)$ be related in such a way that: \begin{equation} [j \in J][k \in K(j)] = [k \in K'][j \in J'(k)] \end{equation}
My understanding of this, is that the set $K'$ is basically the "bounds" that $k$ has, sort of like its restrictions. For the first sum I wrote, I would think that K' be the set {j,j+1,...,n} and since j starts at 1, K' is really just the set of the values $1 \rightarrow n$. However, I can't really figure out $J'(k)$ here , or even if my understanding of $K'$ is right. Can anyone give me some pointers or put me in the right track as to understanding this set relation?
Thanks,