Is there an established notation for sorting finite sequences/words? At the moment, I have written the following definition:
For a word $\textbf{a} = a_1, \dots, a_t$, let $\operatorname{Sort}(\textbf{a}) := a_{\sigma(1)}, \dots, a_{\sigma(t)}$ where $\sigma$ is a permutation such that $a_{\sigma(1)}\le a_{\sigma(2)} \le \dotsb\le a_{\sigma(t)}$.
This is for combinatorics paper I intend to submit to the Arxiv. I am also a little self-conscious that the inclusion of permutations is unnecessary and may cause confusion, so any feedback about that is also welcome.
Note: $a_i$ are all integers so this is all well-defined, just a little verbose
Using words (with a non standard notation) is not a good idea in this case. What you are really doing is the following: you start with a finite sequence $S$ of integers and you produce the sequence $\text{Sort}(S)$ of elements of $S$ in increasing order. For instance, if $S = (2, -1, 0, -1, 3, 2)$, then $\text{Sort}(S)$ is the sequence $(-1, -1,0,2,2,3)$. As you can see, permutations are not needed.
Your notation $\text{Sort}(S)$ makes sense and I would keep it.