On fixed points of permutations

Authors:
Persi Diaconis;Jason Fulman;Robert Guralnick
Affiliations:
Department of Mathematics and Statistics, Stanford, USA 94305;Department of Mathematics, University of Southern California, Los Angeles, USA 90089-2532;Department of Mathematics, University of Southern California, Los Angeles, USA 90089-2532
Venue:
Journal of Algebraic Combinatorics: An International Journal
Year:
2008

Citing 7
Cited 0

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Abstract

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