  We study the natural action of $S_n$ on the set of $k$-subsets of the set
$\{1,\dots, n\}$ when $1\leq k \leq \frac{n}{2}$. For this action we calculate
the maximum size of a minimal base, the height and the maximum length of an
irredundant base.
  Here a "base" is a set with trivial pointwise stabilizer, "height" is the
maximum size of a subset with the property that its pointwise stabilizer is not
equal to the pointwise stabilizer of any proper subset, and an "irredundant
base" can be thought of as a chain of (pointwise) set-stabilizers for which all
containments are proper.
