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Báo cáo toán học: "Records in set partitions"

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:Records in set partitions. | Records in set partitions Arnold Knopfmacher The John Knopfmacher Centre for Applicable Analysis and Number Theory Department of Mathematics University of the Witwatersrand P. O. Wits 2050 Johannesburg South Africa arnold.knopfmacher@wits.ac.za Toufik Mansour Department of Mathematics University of Haifa 31905 Haifa Israel toufik@math.haifa.ac.il Stephan Wagner Department of Mathematical Sciences Stellenbosch University 7602 Stellenbosch South Africa. swagner@sun.ac.za Submitted April 23 2010 Accepted July 13 2010 Published Aug 9 2010 Mathematics Subject Classification Primary 05A18 Secondary 05A15 05A16 Abstract A partition of n 1 2 . n is a decomposition of n into nonempty subsets called blocks. We will make use of the canonical representation of a partition as a word over a finite alphabet known as a restricted growth function. An element ai in such a word n is a strong weak record if ai aj ai aj for all j 1 2 . i 1. Furthermore the position of this record is i. We derive generating functions for the total number of strong weak records in all words corresponding to partitions of n as well as for the sum of the positions of the records. In addition we find the asymptotic mean values and variances for the number and for the sum of positions of strong weak records in all partitions of n . This material is based upon work supported by the National Research Foundation under grant number 2053740 This material is based upon work supported by the National Research Foundation under grant number 70560 THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R109 1 1 Introduction Let n a1a2 an be any permutation of length n. An element ai in n is a record if a aj for all j 1 2 . i 1. Furthermore the position of this record is i. The number of records in permutations was first studied by Renyi 11 see also 5 . A survey of results on this topic can be found in 3 . Recently Myers and Wilf 7 extended the study of records to multiset permutations and words. In the literature records are