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I am looking for some help to simplify the notation of a summation. Let's say we want to sum over a series S with lower and upper bounds of i and N respectively, while excluding a set of indices E. This might be expressed as $$\sum_{\substack{j=i\\j \notin E}}^N S_j$$

or trying to use set builder notation by defining the set of indices as $I = \{k \in \mathbb{Z} : i \leq k \leq N\}$ and then summing:

$$\sum_{\substack{j \in I\\j \notin E}} S_j$$

Now what I want to express is that the upper bound of the summation N increases by the number of indices in I that are contained in E. I might do this by defining another set $H = \{k \in \mathbb{Z} : i \leq k \leq N + |I \in E| \}$ and similar to before:

$$\sum_{\substack{j \in H\\j \notin E}} S_j$$

But this feels quite cumbersome and might not be very good. Is there another way, either with set builder notation or entirely differently, to express that I want to skip indices in a summation and add the amount of skipped indices to the upper bound?

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For the second one, you could also write $$\sum_{j\in I \setminus E} S_j$$

For the third one, you could write $$\sum_{j\in H \setminus E} S_j$$ but $|I \in E|$ should instead be $|I \cap E|$.

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