On fixed points of permutations

Details On fixed points of permutations On fixed points of permutations

2007-08-20

arxiv.org/abs/0708.2643v1

Mathematics

Combinatorics

30 pages

Scientific paper

The number of fixed points of a random permutation of 1,2,...,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,...,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.

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