Notation for binomial coefficient set

Question

I've been searching for a way to express "the set of all combinations generated by taking $\binom{n}{k}$ items". For example, if I have the set $\{3,7,6,5,9\}$, and I want the set of all sets that are formed by making $\binom{5}{4}$ choices, then the result would be

$\{\{3,7,6,5\}, \{3,6,5,9\}, \{3,7,5,9\}, \{3,7,6,9\}, \{7,6,5,9\}\}$

But I'm struggling to find a notation that describes this. I can describe the number of results with $\binom{5}{4}=5$, but how do I describe the resulting set-of-sets itself? The notation $5 \brace 4$ is already taken by Stirling numbers. I've looked at articles and questions about sets, set theory, the binomial coefficient, and I've drawn a blank.

So, is there a standard notation? (I feel that, surely, there must be!) If there is, what is it? And if there isn't, could anyone suggest a notation that would be halfway familiar to a reader?

Answer 1

You want a notation for the set of all $k$-element subsets of some set $X$. There’s a standard notation for this: $[X]^k$. However, it’s most commonly found in set theory and some areas of combinatorics and for a more general audience is quite likely to be unfamiliar to some readers, so you’d do well to define it anyway.

Added: I don’t care for the notation myself, but it occurs to me that I have also seen the notation $\binom{X}k$ used, by analogy with the binomial coefficient itself. Here again it would be a good idea to define the notation if you use it.

Answer 2

• The following notation is used by R.P. Stanley in Topics in Algebraic Combinatorics. See section Basic Notation (p.6)

\begin{array}{ccl} &2^S\qquad&\qquad \text{the set of all subsets of the set } S\\ &\binom{S}{k}\qquad&\qquad \text{the set of $k$-element subsets of } S\\ \end{array}

We can find in section 1.2, p.23 in his classic Enumerative Combinatorics, ed. 2 also the preferred wording for the symbol.

• Now define $\binom{S}{k}$ (sometimes denoted $S^{(k)}$ or otherwise, and read $S$ choose $k$) to be the set of all $k$-element subsets (or $k$-subsets) of $S$ ...