Is there a standard notation for the set of all possible sequences of a set?

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.

• Perhaps refer to these as "sequences" not "series" to be less confusing. Commented Dec 21, 2023 at 10:46
• The set of functions from a prefix of $\{1,2,\ldots,n\}$ to $A$? Commented Dec 21, 2023 at 13:54
• 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... Commented Dec 21, 2023 at 19:05

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 $$[n] \to A$$

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