Abstract
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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This paper is dedicated to the life and work of our colleague Manfred Schocker.
We thank Peter Cameron for his help. Diaconis was supported by NSF grant DMS-0505673. Fulman received funding from NSA grant H98230-05-1-0031 and NSF grant DMS-0503901. Guralnick was supported by NSF grant DMS-0653873. This research was facilitated by a Chaire d’Excellence grant to the University of Nice Sophia-Antipolis.
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Diaconis, P., Fulman, J. & Guralnick, R. On fixed points of permutations. J Algebr Comb 28, 189–218 (2008). https://doi.org/10.1007/s10801-008-0135-2
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DOI: https://doi.org/10.1007/s10801-008-0135-2