Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: On some densities in the set of permutations. | On some densities in the set of permutations Eugenijus Manstavicius Department of Mathematics and Informatics Vilnius University and Institute of Mathematics and Informatics Vilnius Lithuania Submitted Aug 24 2008 Accepted Jun 26 2010 Published Jul 10 2010 Mathematics Subject Classifications 60C05 05A16 Abstract The asymptotic density of random permutations with given properties of the kth shortest cycle length is examined. The approach is based upon the saddle point method applied for appropriate sums of independent random variables. 1 Introduction Let n G N Sn be the symmetric group of permutations acting on the set 1 2 . n and S S 1US2 Set vn for the uniform probability measure on Sn. By vn A vn AnSn we trivially extend it for all subsets of S. If the limit lim vn A d A A c S n exists d A can be called the asymptotic density of A. Let Sc S be the class of A having an asymptotic density d A . The triple S S d is far from being a probability space. However the behavior of d Am for some specialized subsets Am G S as m X are worth to be investigated. In this paper we demonstrate that by taking sets connected to the ordered statistics of different cycle lengths. Recall that each a G Sn can be uniquely up to the order written as a product of independent cycles. Let kj a 0 be the number of cycles of length j 1 j n in such decomposition. The structure vector is defined as k a ki o- . kn a . The final version of this paper was written during the author s stay at Institute of Statistical Science of Academia Sinica Taipei. We gratefully acknowledge the support and thank Professors Hsien-Kuei Hwang and Vytas Zacharovas for the warm hospitality. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R100 1 Set k 1k1 nkn where k k1 . kn G Z then k a n. Moreover if k n then the set a G Sn k a k agrees with the class of conjugate permutations in Sn. If Ệj j 1 are independent Poisson random variables given on some probability space Q F P E j 1 j .
