Double summation, what is the right function? I want to write a summation to count the total number of number of books sold from a list of B different books by P different publishers. That is, there is one list of books and each publisher can sell any of the titles on that list. At any given time, each publisher will no doubt have sold a different number of any given book.
I wonder what would be the correct and understandable notation inside the summation? b = single book, p = single publisher.
$$\sum_{p \in P}\sum_{k=0}^B(p,b_k)$$
$$\sum_{i=0}^P\sum_{k=0}^B(p_i(b_k))$$
$$\sum_{p \in P}\sum_{k=0}^B(f(p,b_k))$$
 A: It seems a little weird to index each book sold if it only counts as $1$ book anyhow. If at all possible, it would be easier to read if written with one summation.
What sounds more plausible is to have a function $f:P\to \Bbb N$ which gives the number of books each publisher sold, and then talk about $\sum_{p\in P}f(p)$. But if you really insist, you could write $f(p)=\sum_{b\in B_p}1$ where $B_p\subset B$ is the subset of books that publisher $p$ sold, and get $\sum_{p\in P}\sum_{b\in B_p}1$.
Summing over books makes more sense if the feature you're asking about differs between books. Say, for example, if each book had a price, and you were summing up the prices of all books from all publishers. Then it would make sense to have some sort of price function $f:P\times B\to \$[0,\infty)$ and then compute $\sum_{p\in P}\sum_{b\in B} f(p,b)$.
A: The problem as now stated has two dimensions. Each publisher has a different set of sales figures of the same set of books. Here, $n(p_j,b_k)$ is the number of copies of title $b_k$ sold by publisher $p_j$.
$$\sum_{j=1}^P\sum_{k=1}^Bn(p_j,b_k)$$
I believe this would make more sense if these were different bookshops rather than publishers. Publishers would be me more likely to have different books on their lists according to ownership of publishing rights. Then, each publisher's sales figures could be summed across their own list and then these individual totals could be summed, but not using the same indexes, which would not be like a typical double sum.
