I'm reading through a paper and it has a summation notation I'm not familiar with:
$$\sum_{n_j;\sum_a n_a = N}$$
how do I read this?
The full line is
$$\mathbb{E}\left[{n_i \choose 2}\right] = \sum_k {k \choose 2} \sum_{n_j;\sum_a n_a = N} \delta(n_i, k) N! \prod_{j=1}^G \frac{p_{j}^{n_j}}{n_j!}$$
The more complete context:
There are $N$ record, and we choose a rule $r$ to group the records into $G(N, r)$ distinct groups. We need to compare the records within each group; call $n_i$ the size of the $i$th group. The total number of [pairwise] comparisons is
$$C(r, N) = \sum_i {n_i \choose 2}$$
...
We assume that the number of records in group $i$ follows the multinomial distribution:
$$P(n_1, n_2, ..., n_G|N) = N! \prod_{i=1}^G \frac{p_{i}^{n_i}}{n_i!}$$
The expected number of [pairwise] comparisons is:
$$\mathbb{E}\left[C(r,N)\right]=\mathbb{E}\left[\sum_i{n_i \choose 2}\right] = \sum_i\mathbb{E}\left[{n_i \choose 2}\right]$$
plugging in the multinomial we get:
$$\mathbb{E}\left[{n_i \choose 2}\right] = \sum_k {k \choose 2} \sum_{n_j;\sum_a n_a = N} \delta(n_i, k) N! \prod_{j=1}^G \frac{p_{j}^{n_j}}{n_j!}$$