For a set $S$ with $n$ elements, the notation for a combination $\binom{n}{k}$, or $C(n, k)$, indicates the number of combinations of $k$ elements from $S$, but how does one indicate the actual set created from combinations of $k$ elements from $S$? That is, $\binom{n}{k}$ is the size of the set I'd like to represent.
Likewise, how would one indicate the actual set of items created from the permutations of $k$ elements, rather than the size of that set?
Wikipedia says that the set of all $k$-combinations of a set $S$ is sometimes denoted by $${S \choose k}$$
The set of combinations is the collection of all subsets of $\{1,2,\ldots,n\}$ of size $k$; if we let $[n]=\{1,2,\ldots,n\}$ (more or less common, depending on the context) $\mathcal{P}(X)$ denote the set of all subsets of $X$, then you are looking for the set $$\bigl\{ A\in\mathcal{P}([n])\bigm| |A|=k\bigr\}.$$ I do not think there is any particular notation for it, but $$\mathcal{P}_k([n])$$ seems reasonable enough. You would have to specify it, though.
For permutations, the order matters. So you are looking for the set of all function $f\colon[k]\to[n]$ that are one-to-one. Again, there is no standard notation, but the set of all functions is $[n]^{[k]}$, so you would want $$\bigl\{ f\in [n]^{[k]}\bigm| f\text{ is one-to-one}\bigr\}.$$ Equivalently, you would want all $k$-tuples that have $k$-distinct elements. So, using the previous notation for subsets of size $k$, you would have: $$\bigl\{ f\in[n]^{[k]}\bigm| f([k]) \in \mathcal{P}_k([n])\bigr\}.$$
A common notation for the collection of all size-$k$ subsets of $S$ is given by the symbol $S_k$.
