Notation for ordered lists with no repetition of elements

Question

If $\{a_1, \ldots, a_n\}$ is the notation for a set of $n$ elements (with no repetition) and $(a_1, \ldots, a_n)$ for an ordered list of $n$ elements (with potential repetition), what is the notation for an ordered list of $n$ elements with no repetition of elements?

Answer 1

Some solutions:

1. Just use textual description, for example

   Let $(a_1,\ldots,a_n)$ be an ordered list of $n$ pairwise distinct elements.

   It doesn't matter that some other lists use the same notation, it's enough to remind the reader sometimes that it was defined this way:

   Note that we can use property (12.34) because elements of sequence $(a_k)_{k=1}^{n}$ were defined to be pairwise distinct (cf. definition 5.67).

2. Introduce your own notation.

   In this paper we will use notation $(\!(a_1, \ldots, a_n)\!)$ to say that elements of sequence $(\!(a_k)\!)_{k=1}^{n}$ are pairwise distinct.

   If you are using $\LaTeX$, then often it is nice to define your own command for that.

3. Use a function instead of a sequence, namely observe that a sequence $(a_1,\ldots,a_n)$ could be defined/manipulated as a function: $f(k) = a_k$. The relevant terms for functions are: injection, injective and one-to-one.

I hope this helps $\ddot\smile$