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?
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$\begingroup$ The set of $k$-tuples from $A$ is simply $A^k$. $\endgroup$– Brian M. ScottDec 11, 2014 at 20:47
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$\begingroup$ Seriously??? That, I did not know... $\endgroup$– JorenHeitDec 11, 2014 at 20:50
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$\begingroup$ Seriously. The Cartesian product of $k$ copies of $A$ is exactly the set of $k$-tuples of elements of $A$. $\endgroup$– Brian M. ScottDec 11, 2014 at 20:52
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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.
