Correct notation or operator to remove elements from sequence

I don't think there's a truly general standard notation. For the removal of single elements, I have sometimes seen the following notation:

$$(s_0, \ldots, \widehat{s_i}, \ldots, s_n),$$ meaning that the $i$-th element was removed.

The most general way would be to use sub-sequences: Define a sequence $i: \mathbb N \to \mathbb N$ that maps to the indices you want to keep, and write $s_{i_j}$.