Ordered cycle lengths in a random permutation

Shepp, Larry; Lloyd, S. P.

doi:10.1090/s0002-9947-1966-0195117-8

Cited by 210 publications

(155 citation statements)

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“…Knuth and Trabb Pardo [39] observe an interesting connection with the distribution of cycle lengths in a random permutation [33,67]. In fact, the distribution of the number of digits of the j-th largest prime factors of n-digit random integers is asymptotically the same as the distribution of lengths of the j-th longest cycles in random permutations of n objects.…”

Section: Performance Of Ecmmentioning

confidence: 97%

Factorization of the tenth Fermat number

Brent

1999

Math. Comp.

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Abstract. We describe the complete factorization of the tenth Fermat number F 10 by the elliptic curve method (ECM). F 10 is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40-digit factor was found after about 140 Mflop-years of computation. We also discuss the complete factorization of other Fermat numbers by ECM, and summarize the factorizations of F 5 , . . . , F 11 .

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Section: Performance Of Ecmmentioning

confidence: 97%

Factorization of the tenth Fermat number

Brent

1999

Math. Comp.

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“…The author also wishes to thank M. Steele for his interest in this work and his reference to [2], [7], and [10]. My conversations with M. Steele have been very interesting and productive.…”

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confidence: 92%

“…In Section 3 we calculate an auxiliary quantity which is used in Section 4 to prove (i). In Section 5 we use random permutations and the remarkable asymptotic results of Shepp and Lloyd [7] and Vershik and Shmidt [12] to prove (ii). In Section 6 we prove (iii) and discover a connection with the Stirling numbers of the first kind.…”

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confidence: 99%

“…We begin this sec tion by recalling the remarkable results of Shepp and Lloyd [7] and Vershik and Shmidt [12] on the asymptotic distributions of cycle lengths of ran dom permutations. Denote the symmetric group on n symbols by 5P n and endow it with the uniform probability distribution P. For 7Г £ SP n , de fine а г (7г), i -1, 2,..., тг, to be the number of cycles of 7Г of length i; we refer to (ai(7r),.. .,a n (7r)) as the cycle type of 7Г.…”

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confidence: 99%

“…Actually, the connection between the two objects has a history which is given in the Goldie's paper [4]. Shepp and Lloyd [7] proved that the limit distribution of /n can be explicitly described by the following distribution function We can say even more. Vershik and Shmidt [12] …”

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confidence: 99%

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Convex Minorants of Random Walks and Brownian Motion

Suidan¹

2001

Teor. Veroyatnost. i Primenen.

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CONVEX MINORANTS OF RANDOM WALKS AND BROWNIAN MOTIONПусть (S,-)JL 0 -процесс случайного блуждания, порожденный последовательностью независимых и одинаково распределенных ве-щественнозначных случайных величин (Х,)" =1 , имеющих плот ность. Изучаются вероятностные распределения, связанные с ас социированным процессом выпуклой миноранты. В частности, ис следуется длина самого длинного сегмента выпуклой миноранты. Используя теорию случайных перестановок, мы полностью харак теризуем распределение длины r-го по величине сегмента выпуклой миноранты броуновского движения на конечных интервалах; мы также указываем явный вид плотности совместного распределения г первых по длине сегментов. Кроме того, мы используем развитые здесь методы для доказательства формулы (Е. Sparre Andersen, [9]), позволяющей вычислить вероятность того, что выпуклая миноран та случайного блуждания длины N будет состоять из т сегментов. Приводятся аналогичные утверждения для случайных блужданий со случайными приращениями времени. Эти результаты недавно использованы автором для изучения динамики одномерных частиц с прилипанием.Ключевые слова и фразы: случайное блуждание, выпуклая ми норанта, броуновское движение, случайные перестановки. 1.Introduction. In this paper we study properties of convex minorants of random walks generated by real-valued independent identically dis tributed (i.i. (i) For a random walk of length TV, the probability that the associated convex minorant contains a segment of length n is n~l for n > N/2. The expected number of segments of length n is n~l.(ii) For s G [0,1] and any positive integer r, the probability that the length of the r-th longest segment, i^, of the convex minorant generated by Brownian motion on [0,1] is less than s is F r (s), where F r (s) has an explicit formula; the joint density for the lengths of the r longest segments,

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On the multiplicity of parts in a random partition

Corteel

Pittel

Savage

et al. 1999

Random Struct. Alg.

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Let λ be a partition of an integer n chosen uniformly at random among all such partitions. Let s(λ) be a part size chosen uniformly at random from the set of all part sizes that occur in λ. We prove that, for every fixed m≥1, the probability that s(λ) has multiplicity m in λ approaches 1/(m(m+1)) as n→∞. Thus, for example, the limiting probability that a random part size in a random partition is unrepeated is 1/2. In addition, (a) for the average number of different part sizes, we refine an asymptotic estimate given by Wilf, (b) we derive an asymptotic estimate of the average number of parts of given multiplicity m, and (c) we show that the expected multiplicity of a randomly chosen part size of a random partition of n is asymptotic to (log n)/2. The proofs of the main result and of (c) use a conditioning device of Fristedt. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14, 185–197, 1999

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Ordered cycle lengths in a random permutation

Cited by 210 publications

References 2 publications

Factorization of the tenth Fermat number

Factorization of the tenth Fermat number

Convex Minorants of Random Walks and Brownian Motion

On the multiplicity of parts in a random partition

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