Is there an operator that generates a set of combinations from a set?

Question

Example Input: $$\mathbb{S} = \{1,2,3\}$$

Proposed Operator: $$\mathbb{C} = \operatorname{K}(\mathbb{S},k=2)$$

Example Output: $$\mathbb{C} = \Big\{\{1,2\},\{2,3\},\{1,3\}\Big\}$$

Side note: the operator $C^{n}_{k}$ for finding the number of $k$-size combinations in $\mathbb{S}$ itself is surprisingly common.

Answer 1

The "operator" is simply called the set of $k$-subsets of $S$. From the literature, I know of at least four different notations for it: $$ \mathcal{P}_k(S), \space [S]^k, \space \binom{S}{k} \space \text{and} \space S^{(k)}. $$ Choose your favorite one!

Answer 2

Something I've commonly seen written is $$\binom{\mathbb S}2$$ This obviously generalizes to $n$-element sets. When the top element in the binomial symbol is a set, there's really only one thing you can mean.