On the Maximal Multiplicity of Parts in a Random Integer Partition

Mutafchiev, Ljuben

doi:10.1007/s11139-005-1870-9

Cited by 11 publications

(22 citation statements)

References 13 publications

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“…(We refer the reader e.g. to [23], [24], [25], [6], [7], [18], [11], [15] and the references therein.) To this effect, Theorem 1 continues the study of partitions initiated by the probabilistic approach.…”

Section: Introductionmentioning

confidence: 99%

The limiting distribution of the trace of a random plane partition

Kamenov

Mutafchiev

2007

Acta Math Hung

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Section: Introductionmentioning

confidence: 99%

The limiting distribution of the trace of a random plane partition

Kamenov

Mutafchiev

2007

Acta Math Hung

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“…Recently, Brennan, Knopfmacher and the author [3] showed that the number of ascents of at least a given size d follows a Gaussian law. The longest run (or largest ascent), however, has a quite unusual and interesting distribution, as recently shown by Mutafchiev [14]-the corresponding distribution function is closely related to the function that generates partitions. Finally, another recent paper by Grabner and Knopfmacher [9] that discusses various partition statistics based on gaps deserves to be mentioned.…”

Section: Introductionmentioning

confidence: 72%

“…It was noted in Mutafchiev's paper [14] that Theorem 5 implies weak convergence of the distribution of the normalized random variable "longest run in a random partition of n" to the distribution of the random variable…”

Section: Discussionmentioning

confidence: 99%

“…The saddle point method is the most common way to treat problems of such a type, see for instance [19] or [14]. It starts with the observation that we have…”

Section: The General Saddle Point Approachmentioning

confidence: 99%

“…Such estimates are well known for f (see e.g. [14] and the references therein), but the additional term −f (dt) can make things a little more intricate; for the sake of completeness, we include the following lemma with a short proof: Lemma 2. Assume that there are constants c 1 and c 2 such that…”

Section: The General Saddle Point Approachmentioning

confidence: 99%

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On the distribution of the longest run in number partitions

Wagner

2009

Ramanujan J

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Abstract. We consider the distribution of the longest run of equal elements in number partitions (equivalently, the distribution of the largest gap between subsequent elements); in a recent paper, Mutafchiev proved that the distribution of this random variable (appropriately rescaled) converges weakly. The corresponding distribution function is closely related to the generating function for number partitions. In this paper, this problem is considered in more detail-we study the behavior at the tails (especially the case that the longest run is comparatively small) and extend the asymptotics for the distribution function to the entire interval of possible values. Additionally, we prove a local limit theorem within a suitable region, i.e. when the longest run attains its typical order n 1/2 , and we observe another phase transition that occurs when the largest gap is of order n 1/4 : there, the conditional probability that the longest run has length d, given that it is ≤ d, jumps from 1 to 0. Asymptotics for the mean and variance follow immediately from our considerations.

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On the maximal multiplicity of block sizes in a random set partition

Mutafchiev

Savov

2019

Random Struct Algorithms

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We study the asymptotic behavior of the maximal multiplicity Mn = Mn(σ) of the block sizes in a set partition σ of [n] = {1,2,…,n}, assuming that σ is chosen uniformly at random from the set of all such partitions. It is known that, for large n, the blocks of a random set partition are typically of size W = W(n), with WeW = n. We show that, over subsequences {nk}k ≥ 1 of the sequence of the natural numbers, Mnk, appropriately normalized, converges weakly, as k→∞, to maxfalse{Z1,Z2−ufalse}, where Z1 and Z2 are independent copies of a standard normal random variable. The subsequences {nk}k ≥ 1, where the weak convergence is observed, and the quantity u depend on the fractional part fn of the function W(n). In particular, we establish that limk→∞1false/false(2πfalse)1false/4minfalse{fnk,1−fnkfalse}nkfalse/normallog7false/4nk=u∈false[0,∞false)∪false{∞false}. The behavior of the largest multiplicity Mn is in a striking contrast to the similar statistic of integer partitions of n. A heuristic explanation of this phenomenon is also given.

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On the Maximal Multiplicity of Parts in a Random Integer Partition

Cited by 11 publications

References 13 publications

The limiting distribution of the trace of a random plane partition

The limiting distribution of the trace of a random plane partition

On the distribution of the longest run in number partitions

On the maximal multiplicity of block sizes in a random set partition

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