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Báo cáo toán học: " Derangements and Euler’s difference table for C"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Derangements and Euler’s difference table for C. | Derangements and Euler s difference table for C o Sn Hilarion L. M. Faliharimalala1 and Jiang Zeng2 1 Departement de Mathematiques et Informatique Universite d Antananarivo 101 Antananarivo Madagascar hilarion@ist-tana.mg 2Universite de Lyon Universite Lyon 1 Institut Camille Jordan UMR 5208 du CNRS F-69622 Villeurbanne Cedex France zeng@math.univ-lyon1.fr Submitted Dec 26 2007 Accepted Apr 22 2008 Published Apr 28 2008 Mathematics Subject Classifications 05A18 05A15 05A30 Abstract Euler s difference table associated to the sequence n leads naturally to the counting formula for the derangements. In this paper we study Euler s difference table associated to the sequence nn and the generalized derangement problem. For the coefficients appearing in the later table we will give the combinatorial interpretations in terms of two kinds of k-successions of the group C o Sn. In particular for 1 we recover the known results for the symmetric groups while for 2 we obtain the corresponding results for the hyperoctahedral groups. 1 Introduction The probleme de rencontres in classical combinatorics consists in counting permutations without fixed points see 6 p. 9-12 . On the other hand one finds in the works of Euler see 11 the following table of differences n n and gn Un - g -i 0 m n - 1 Clearly this table leads naturally to an explicit formula for gn which corresponds to the number of derangements of n 1 n . As n is the cardinality of the symmetric group of n Euler s difference table can be considered to be an array associated to the symmetric group. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R65 1 In the last two decades much effort has been made to extend various enumerative results on symmetric groups to other Coxeter groups the wreath product of a cyclic group with a symmetric group and more generally to complex reflection groups. The reader is referred to 1 2 10 9 12 14 4 5 3 and the references cited there for the recent works in this direction. In this paper we .