Geodesic
Ideals and filters of partitions and cyclic classes, and invariance domains of permutations
Diskretnaya Matematika, Tome 8 (1996) no. 3, pp. 3-21 
V. N. Sachkov. Ideals and filters of partitions and cyclic classes, and invariance domains of permutations. Diskretnaya Matematika, Tome 8 (1996) no. 3, pp. 3-21. http://geodesic.mathdoc.fr/item/DM_1996_8_3_a0/
@article{DM_1996_8_3_a0,
     author = {V. N. Sachkov},
     title = {Ideals and filters of partitions and cyclic classes, and invariance domains of permutations},
     journal = {Diskretnaya Matematika},
     pages = {3--21},
     year = {1996},
     volume = {8},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_1996_8_3_a0/}
}
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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 $$ P(n,k)\leq 1-e^{-1}+o(1). $$