  We prove that the symmetric group $S_n$ has a unique minimal cover
$\mathcal{M}$ by maximal nilpotent subgroups, and we obtain an explicit and
easily computed formula for the order of $\mathcal{M}$. In addition, we prove
that the order of $\mathcal{M}$ is equal to the order of a maximal
non-nilpotent subset of $S_n$. This cover $\mathcal{M}$ has attractive
properties; for instance, it is a normal cover, and the number of conjugacy
classes of subgroups in the cover is equal to the number of partitions of $n$
into distinct positive integers. We show that these results contrast with those
for the alternating group $A_n$. In particular, we prove that, for all but
finitely many values of $n$, no minimal cover of $A_n$ by maximal nilpotent
subgroups is a normal cover and the order of a minimal cover of $A_n$ by
maximal nilpotent subgroups is strictly greater than the order of a maximal
non-nilpotent subset of $A_n$.
