Geodesic
On fixed points of permutations.
Journal of Algebraic Combinatorics, Tome 28 (2008) no. 1, pp. 189-218.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: The number of fixed points of a random permutation of ${1,2,\cdots , n}$ has a limiting Poisson distribution. We seek a generalization, looking at other actions of the symmetric group. Restricting attention to primitive actions, a complete classification of the limiting distributions is given. For most examples, they are trivial - almost every permutation has no fixed points. For the usual action of the symmetric group on $k$-sets of ${1,2,\cdots , n}$, the limit is a polynomial in independent Poisson variables. This exhausts all cases. We obtain asymptotic estimates in some examples, and give a survey of related results.
Keywords: keywords fixed point, derangement, primitive action, o'nan-Scott theorem 
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Diaconis, Persi; Fulman, Jason; Guralnick, Robert. On fixed points of permutations.. Journal of Algebraic Combinatorics, Tome 28 (2008) no. 1, pp. 189-218. http://geodesic.mathdoc.fr/item/JAC_2008__28_1_a2/