Cyclic permutations of sequences and uniform partitions

Po-Yi Huang, Jun Ma, Yeong Nan Yeh

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5 Citations (Scopus)

Abstract

Let r = (ri)i=1n be a sequence of real numbers of length n with sum s. Let s0 = 0 and si = r1 +. . .+ ri for every i ∈ {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 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 ri = (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.

Original languageEnglish
Pages (from-to)1-12
Number of pages12
JournalElectronic Journal of Combinatorics
Volume17
Issue number1
Publication statusPublished - 2010 Oct 1

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All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Applied Mathematics

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