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Predecessors in Random Mappings

Published online by Cambridge University Press:  12 September 2008

Gerd Baron ,
Michael Drmota  and
Ljuben Mutafchiev
Gerd Baron
Affiliation:
Department of Discrete Mathematics, Technical University of Vienna, Wiedner Hauptstrasse 8–10/118, A-1040 Vienna, Austria
Michael Drmota
Affiliation:
Department of Discrete Mathematics, Technical University of Vienna, Wiedner Hauptstrasse 8–10/118, A-1040 Vienna, Austria
Ljuben Mutafchiev
Affiliation:
Institute of Mathematics, Bulgarian Academy of Sciences, P.O. Box 373, 1090 Sofia, Bulgaria
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Abstract

Let ℱn be the set of random mappings ϕ : {1,…,n} → {1,…,n} (such that every mapping is equally likely). For x ε {l,…,n} the elements are called the predecessors of x. Let Nr denote the random variable which counts the number of points x ε {l,…,n} with exactly r predecessors. In this paper we identify the limiting distribution of Nr as n → ∞. If r = r(n) = o(n⅔) then the limiting distribution is Gaussian, if r ˜ Cn⅔ then it is Poisson, and in the remaining case rn−⅔ → ∞ it is degenerate. Furthermore, it is shown that Nr is a Poisson approximation if r → ∞.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 5 , Issue 4 , December 1996 , pp. 317 - 335
Copyright
Copyright © Cambridge University Press 1996

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