Notation for all k-tuples that can be constructed from a set

Question

Is there a generally accepted notation for a $k$-tuple that is constructed from a set? I have a set $\mathcal{A}$, and need to sum over all possible $k$-tuples (denoted $t_k$). Right now, I'm using set-notation like so: $$ \sum_{t_k\subseteq\mathcal{A}} ... $$ Because the $t_k$ aren't sets, I think this notation is not formally correct. Is there a better alternative?

Answer 1

The cartesian product is your answer. \begin{equation} \mathcal{A}^{k} = \mathcal{A} \times \cdots \times \mathcal{A} = \{ (t_{1},\ldots, t_{k}) \, | \, t_{j} \in \mathcal{A} \textrm{ for } 1 \leq j \leq k \}. \end{equation} In the case that $\mathcal{A} = \mathbb{Z}_{\geq 0}$, such $k$-tuples are often called multi-indices,since they are used in the notation for multivariate partial derivatives.