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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: Generalized Bell polynomials and the combinatorics of Poisson central moments. | Generalized Bell polynomials and the combinatorics of Poisson central moments Nicolas Privault Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University SPMS-MAS 21 Nanyang Link Singapore 637371 nprivault@ntu.edu.sg Submitted Jul 13 2010 Accepted Feb 26 2011 Published Mar 11 2011 Mathematics Subject Classifications 11B73 60E07 Abstract We introduce a family of polynomials that generalizes the Bell polynomials in connection with the combinatorics of the central moments of the Poisson distribution. We show that these polynomials are dual of the Charlier polynomials by the Stirling transform and we study the resulting combinatorial identities for the number of partitions of a set into subsets of size at least 2. 1 Introduction The moments of the Poisson distribution are well-known to be connected to the combinatorics of the Stirling and Bell numbers. In particular the Bell polynomials Bn A satisfy the relation Bn A Ex Zn n G N 1.1 where Z is a Poisson random variable with parameter A 0 and n Bn 1 E S n c 1.2 c 0 is the Bell number of order n i.e. the number of partitions of a set of n elements. In this paper we study the central moments of the Poisson distribution and we show that they can be expressed using the number of partitions of a set into subsets of size at least 2 in connection with an extension of the Bell polynomials. THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P54 1 Consider the above mentioned Bell or Touchard polynomials Bn A defined by the exponential generating function eA e -1 Ể n Bn A . 1.3 n 0 A t G R cf. e.g. 11.7 of 4 and given by the Stirling transform n Bn A J2 AcS n c 1.4 c 0 where 1 c Sin c W-fic l 1 in 15 S n c c 2- 1 ir 1-5 I 0 k denotes the Stirling number of the second kind i.e. the number of ways to partition a set of n objects into c non-empty subsets cf. 1.8 of 7 Proposition 3.1 of 3 or 3.1 of 6 and Relation 1.2 above. In this note we define a two-parameter generalization of the
