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Báo cáo toán học: "Cyclic permutations of sequences and uniform 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: Cyclic permutations of sequences and uniform partitions. | Cyclic permutations of sequences and uniform partitions Po-Yi Huang Department of Mathematics National Cheng Kung University Tainan Taiwan pyhuang@mail.ncku.edu.tw Jun Ma Department of Mathematics Shanghai Jiao Tong University Shanghai China majun904@sjtu.edu.cn Yeong-Nan Yeh Institute of Mathematics Academia Sinica Taipei Taiwan mayeh@math.sinica.edu.tw Submitted Apr 25 2010 Accepted Jul 28 2010 Published Aug 24 2010 Mathematics Subject Classification 05A18 Abstract Let r ri n 1 be a sequence of real numbers of length n with sum s. Let So 0 and Si r1 . Vi for every i G 1 2 . n . Fluctuation theory is the name given to that part of probability theory which deals with the fluctuations of the partial sums si. Define p r to be the number of positive sum si among S1 . sn and m r to be the smallest index i with Si max Sk. An important problem in 0 k n fluctuation theory is that of showing that in a random path the number of steps on the positive half-line has the same distribution as the index where the maximum is attained for the first time. In this paper let fi ri . rn r1 . ri-1 be the i-th cyclic permutation of r. For s 0 we give the necessary and sufficient conditions for m ri 1 i n 1 2 . n and p ri 1 i n 1 2 . n for s 0 we give the necessary and sufficient conditions for m ri 1 i n 0 1 . n 1 and p ri 1 i n 0 1 . n 1 . We also give an analogous result for the class of all permutations of r. Keywords Cyclic permutation Fluctuation theory Uniform partition Partially supported by NSC 96-2115-M-006-012 1 Corresponding author Partially supported by NSC 96-2115-M-001-005 THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R117 1 1 Introduction Fluctuation theory is the name given to that part of probability theory which deals with the fluctuations of the partial sums si xi . xi of a sequence of random variables xi . xn. An important problem in fluctuation theory is that of showing that in a random path the number of steps on the positive half-line has the same distribution as