Ideals and filters of partitions and cyclic classes, and invariance domains of permutations

We give explicit formulae for the probability $P(n,k)$ that the random equiprobable permutation of degree $n$ has an invariant $k$-subset, $1\leq k\leq n/2$, and their asymptotic representations are found for any fixed $k$ as $n\to\infty$. It is shown that under these conditions
$$ P(n,k)\leq 1-k\exp\left\{-\sum_{j=1}^k {1\over j}\right\}+o(1), $$
and hence
$$ P(n,k)\leq 1-e^{-1}+o(1). $$