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Is there a name/notation for the set of n-tuples of elements of some set? Like $S_n$ for permutations, but with replacement.

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Typically, $S^n = S \times S \times \cdots \times S = \{(s_1,s_2,\ldots,s_n) : s_j \in S \mbox{ for } 1 \le j \le n\}$.

Is that what you mean?

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    This is often referred to as a Cartesian product (or, in this case, $n-1$ such products). – 2017-02-27
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    Yes, this was it. I derped hard, thanks for the help. – 2017-02-27