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@ Jun Ma Department of Mathematics Shanghai Jiao Tong University Shanghai China majun904@ Yeong-Nan Yeh Institute of Mathematics Academia Sinica Taipei Taiwan mayeh@ 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
