I am not aware of any such notation (and in the business of choice functions, one runs a lot into $\mathcal P(S)\setminus\{\varnothing\}$).
It is fine to make your own, but be sure to be consistent about it, and to define it at the beginning of your work.
There is a risk of having too many notations, it may burden the reader. Sometimes just writing it explicitly works just as well. If you're tired of doing that, write a LaTeX macro.
Sets are collections of mathematical objects without importance to the order or repetition. That is $$\{0,0,0,0,0,0,1,1,1,1,2,2,2,2,1,1,1,1,0\}=\{0,1,2\}=\{2,2,1,0\}=\ldots$$
If you are interested in the order then you wish to talk about sequences rather than sets. Sequences are often denoted by $\langle a_i\mid i\in I\rangle$ where $I$ is an index set which carries (usually) some natural order, at least in the case of sequences. For example $I$ can be taken as the natural numbers or a finite subset of them. If the index set is very small we can just write the sequences as $\langle a_1,\ldots,a_n\rangle$.
So we have $\langle 21,34,42\rangle$. We can treat this as a function from $\{1,2,3\}$ into some other set, that is $h(1)=21, h(2)=34, h(3)=42$. Then we can write $h(1)$ or $h_1$ for the first element of the sequence.
Best Answer
The notation $\Bbb A - \{a\}$ is often used to mean the same thing as $\Bbb A \setminus \{a\}$ (the set difference), but I've never seen it with a tilde and can't find any references to it being used this way with Google.
The tilde $\sim$ is sometimes used as a negation or "not" symbol in set theory, in which case
$$\Bbb A \setminus \{a\} = \bigl\{x : x \in \Bbb A, \sim\!(x\in\{a\})\bigr\}.$$
The tilde is also used sometimes for equivalence relations, where $x \sim y$ means $x$ and $y$ are equivalent (in the same equivalence class) under some equivalence relation $\sim$.
A particularly common example of this is with the cardinality of sets. We say $A \sim B$ if $A$ and $B$ have the same cardinality, that is $|A| = |B|$, and we call them equinumerous.