On some problems of the statistical theory of partitions with application to characters of the symmetric group. II

Szalay, Mihály; Túrán, P.

doi:10.1007/bf01895858

Cited by 31 publications

(25 citation statements)

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“…Erdös and Lehner [14] introduced a probabilistic viewpoint that was quite fruitful. Random partitions were developed by Erdös, Szalay, Turan and others, [15,32,[34][35][36][37]. Szekeres in [38] studied the joint distribution of the number of parts and the maximal part size in integer partitions.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Random partitions with restricted part sizes

Goh¹,

Hitczenko²

2007

Random Struct Algorithms

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ABSTRACT:For a subset S of positive integers let (n, S) be the set of partitions of n into summands that are elements of S. For every λ ∈ (n, S), let M n (λ) be the number of parts, with multiplicity, that λ has. Put a uniform probability distribution on (n, S), and regard M n as a random variable. In this paper the limiting density of the (suitably normalized) random variable M n is determined for sets that are sufficiently regular. In particular, our results cover the case S = {Q(k) : k ≥ 1}, where Q(x) is a fixed polynomial of degree d ≥ 2. For specific choices of Q, the limiting density has appeared before in rather different contexts such as Kingman's coalescent, and processes associated with the maxima of Brownian bridge and Brownian meander processes.

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Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Random partitions with restricted part sizes

Goh¹,

Hitczenko²

2007

Random Struct Algorithms

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“…In [1] and [2] Dartyge and Sárközy studied the distribution of the summands of (unrestricted) partitions in residue classes and, in particular, they showed that if d ∈ N * is fixed and n → +∞, then the summands of almost all partitions of n are well-distributed modulo d. Moreover, they applied this result to study the rate of the square-free summands to all summands in a "random" partition of n. (See [4] and [6] for further related results. )…”

Section: Introduction and Resultsmentioning

confidence: 98%

On the distribution of the summands of partitions in residue classes

Dartyge

Sárközy

Szalay³

2005

Acta Math Hung

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“…Many years later Szalay and Turán [9][10][11] undertook a rigorous study of the uniformly random integer partition, and proved a central limit theorem for the moderately sized parts. Vershik [13] noticed that a weak law following from Szalay-Turán's result does indeed imply the above equation.…”

Section: Statements and Proofsmentioning

confidence: 99%

Limit shape of a random integer partition with a bounded max-to-min ratio of parts sizes

Pittel

2007

Journal of Combinatorial Theory, Series A

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We consider an integer partition λ 1 · · · λ , 1, chosen uniformly at random among all partitions of n such that λ 1 /λ does not exceed a given number k > 1. For k = 2, Igor Pak had conjectured existence of a constant a such that the random function m −1 n λ xm n , x ∈ [0, 1] (m n = an 1/2 ), converges in probability to y = f (x) 1, f (0) = 2, f (1) = 1, whose graph is symmetric with respect to y = x + 1. We confirm a natural extension of Pak's conjecture for k > 1, and show that the limit shape y = f (x) is given by w x+1 + w y = 1, where w k + w = 1. In particular, for k = 2, w is the golden ratio ( √ 5 − 1)/2.

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On some problems of the statistical theory of partitions with application to characters of the symmetric group. II

Cited by 31 publications

References 0 publications

Random partitions with restricted part sizes

Random partitions with restricted part sizes

On the distribution of the summands of partitions in residue classes

Limit shape of a random integer partition with a bounded max-to-min ratio of parts sizes

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