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The standard notation for the set of all subsets (power set) of a set $A$ is either $\wp(A)$ or $2^A$.

I'd like to know whether there's a standard notation and name for the set of all sequences of length from $1$ to $n$ that can be formed from a set $A$, where $n$ is the cardinality of $A$.

That is, where $A = \{a_1, a_2, ..., a_n\}$, I want to know if there's a standard name and notation for the set:

$$\{(a_1), (a_2), ..., (a_1, a_1, a_1), ..., (a_4, a_5, a_1), ..., (a_2, a_1, a_3, a_3), ..., (a_n, ..., a_{n-1}), (a_n, ..., a_n)\},$$

where the last two sequences have $n$ terms.

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    $\begingroup$ Perhaps refer to these as "sequences" not "series" to be less confusing. $\endgroup$
    – GEdgar
    Commented Dec 21, 2023 at 10:46
  • $\begingroup$ The set of functions from a prefix of $\{1,2,\ldots,n\}$ to $A$? $\endgroup$
    – vincent163
    Commented Dec 21, 2023 at 13:54
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    $\begingroup$ Also, as you probably know, but might want to think about in a careful formulation of the question, that if, for example, $a_1=a_2$, then the apparent listing of elements of the set $A$ "collapses". E.g., $\{1,1,1,2\}=\{1,2\}$. This makes the details of the question potentially a little ambiguous... $\endgroup$ Commented Dec 21, 2023 at 19:05

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There is no standard notation for this exact construction that I am aware of, but it can be written rather succinctly still (btw, I'm assuming you mean sequence and not series) as

$$ \bigcup_{i=1}^{n} A^{[i]} $$

If you have not seen this notation before

$$ A^X $$

Denotes the set of functions from $X$ to $A$ (this is motivated by $n^m = |[n]^{[m]}|$ )

And a sequence in $A$ of length $n$ is formally defined as a function from $[n] \to A$

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  • $\begingroup$ You can write $A^i$ directly, instead of $A^{[i]}$, for the "Cartesian power" of A. $\endgroup$
    – Stef
    Commented Dec 21, 2023 at 18:55

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