{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:54Z","timestamp":1753893834447,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $\\vec{r}=(r_i)_{i=1}^n$ be a sequence of  real numbers of length $n$ with sum $s$. Let $s_0=0$ and $s_i=r_1+\\ldots +r_i$ for every $i\\in\\{1,2,\\ldots,n\\}$. Fluctuation theory is the name given to that part of probability theory which deals with the fluctuations of the partial sums $s_i$. Define $p(\\vec{r})$ to be the number of positive sum $s_i$ among $s_1,\\ldots,s_n$  and $m(\\vec{r})$ to be the smallest index $i$ with $s_i=\\max\\limits_{0\\leq k\\leq n}s_k$. 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 $\\vec{r}_i=(r_i,\\ldots,r_n,r_1,\\ldots,r_{i-1})$ be the $i$-th cyclic permutation of $\\vec{r}$. For $s&gt;0$, we give the necessary and sufficient conditions for $\\{ m(\\vec{r}_i)\\mid 1\\leq i\\leq n\\}=\\{1,2,\\ldots,n\\}$ and $\\{ p(\\vec{r}_i)\\mid 1\\leq i\\leq n\\}=\\{1,2,\\ldots,n\\}$; for $s\\leq 0$, we give the necessary and sufficient conditions for $\\{ m(\\vec{r}_i)\\mid 1\\leq i\\leq n\\}=\\{0,1,\\ldots,n-1\\}$ and $\\{ p(\\vec{r}_i)\\mid 1\\leq i\\leq n\\}=\\{0,1,\\ldots,n-1\\}$. We also give an analogous result for the class of all permutations of $\\vec{r}$.<\/jats:p>","DOI":"10.37236\/389","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:10:22Z","timestamp":1578715822000},"source":"Crossref","is-referenced-by-count":4,"title":["Cyclic Permutations of Sequences and Uniform Partitions"],"prefix":"10.37236","volume":"17","author":[{"given":"Po-Yi","family":"Huang","sequence":"first","affiliation":[]},{"given":"Jun","family":"Ma","sequence":"additional","affiliation":[]},{"given":"Yeong-Nan","family":"Yeh","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2010,8,24]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r117\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r117\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:27:46Z","timestamp":1579303666000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v17i1r117"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,8,24]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2010,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/389","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2010,8,24]]},"article-number":"R117"}}