On the multiplicity of parts in a random partition

Corteel, Sylvie; Pittel, Boris; Savage, Carla D.; Wilf, Herbert S.

doi:10.1002/(sici)1098-2418(199903)14:2<185::aid-rsa4>3.0.co;2-f

Cited by 37 publications

(42 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The asymptotic behaviour of the mean was found by Corteel, Pittel, Savage and Wilf [4]: the average number of parts of multiplicity d is asymptotically √ 6n/(πd(d + 1)). The limiting distribution is Gaussian for fixed d: see [19].…”

Section: Number Of Parts With Given Multiplicitymentioning

confidence: 76%

“…(see [4,Theorem 1]), so that the generating function for the total number of parts of multiplicity d is given by…”

Section: Number Of Parts With Given Multiplicitymentioning

confidence: 99%

“…and to use Rademacher's formula (1.2) to approximate p(k) in this sum, so that it can be determined by means of the Euler-Maclaurin formula or other techniques (see for instance [3] or [4]). The second possibility is to work directly with the generating function and to apply Cauchy's integral formula (as for instance in [13] or [14]) in combination with the saddle point method.…”

Section: Longest Runmentioning

confidence: 99%

See 2 more Smart Citations

A General Asymptotic Scheme for the Analysis of Partition Statistics

Grabner¹,

Knopfmacher

Wagner

2014

Combinator. Probab. Comp.

View full text Add to dashboard Cite

We consider statistical properties of random integer partitions. In order to compute means, variances and higher moments of various partition statistics, one often has to study generating functions of the form P (x)F(x), where P (x) is the generating function for the number of partitions. In this paper, we show how asymptotic expansions can be obtained in a quasi-automatic way from expansions of F(x) around x = 1, which parallels the classical singularity analysis of Flajolet and Odlyzko in many ways. Numerous examples from the literature, as well as some new statistics, are treated via this methodology. In addition, we show how to compute further terms in the asymptotic expansions of previously studied partition statistics.

show abstract

Section: Number Of Parts With Given Multiplicitymentioning

confidence: 76%

“…(see [4,Theorem 1]), so that the generating function for the total number of parts of multiplicity d is given by…”

Section: Number Of Parts With Given Multiplicitymentioning

confidence: 99%

Section: Longest Runmentioning

confidence: 99%

See 1 more Smart Citation

A General Asymptotic Scheme for the Analysis of Partition Statistics

Grabner¹,

Knopfmacher

Wagner

2014

Combinator. Probab. Comp.

View full text Add to dashboard Cite

show abstract

“…• Herb considered the following: As n → ∞, what is the probability that a randomly chosen part size in a random partition of n has multiplicity m? It turns out that the probability approaches 1/(m(m + 1)) as n → ∞ [15].…”

Section: Albert Nijenhuismentioning

confidence: 99%

Herbert S. Wilf (1931–2012)

Chung¹,

Greene²,

Hutchinson³

et al. 2015

Notices Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

“…In this paper we consider the multiplicity of a randomly chosen part size in a random composition of an integer n. Let us recall that a multiset = f 1 ; : : : ; k g is a partition of an integer n if the j are positive integers, called parts, such that P j = n. Compositions are merely partitions in which the order of parts is signi cant. Thus, for example, the integer 3 admits three partitions, f1; 1; 1g, f2; 1g and f3g, and four compositions, namely (1; 1; 1), (1; 2), (2; 1) and (3).…”

Section: Introductionmentioning

confidence: 99%

On the Multiplicity of Parts in a Random Composition of a Large Integer

Hitczenko¹,

Savage²

2004

SIAM J. Discrete Math.

Self Cite

View full text Add to dashboard Cite

In this paper we study the following question posed by H. S. Wilf: What is, asymptotically as n ! 1, the probablility that a randomly chosen part size in a random composition of an integer n has multiplicity m ? More speci cally, given positive integers n and m, suppose that a composition of n is selected uniformly at random and then, out of the set of part sizes in , a part size j is chosen uniformly at random. Let P(A (m) n) be the probability that j has multiplicity m. We show that for xed m, P(A (m) n) goes to 0 at the rate 1= ln n. A more careful analysis uncovers an unexpected result: (ln n)P(A (m) n) does not have a limit but instead oscillates around the value 1=m as n ! 1. This work is a counterpart of a recent paper of Corteel, Pittel, Savage and Wilf who studied the same problem in the case of partitions rather than compositions. Integer partitions (as deterministic objects) have been studied for quite some time, but Erd os and Lehner 5] were apparently the rst to study integer partitions from the probabilistic perspective, namely, they considered the set of all partitions, P(n), of an integer n, as a probability space equipped with the uniform probability measure. Quantities of interest are treated as random variables and one can study their probabilistic properties, most typically, the limiting properties as n ! 1. Erd os and Lehner, for example, considered the limiting distribution of the total number of parts in a partition. Their paper opened a new line of investigation. Goh and Schmutz 10] obtained the central limit theorem for the number of di erent part sizes in a random partition, that is, they proved that the number of di erent part sizes, appropriately normalized, has, approximately, the standard Gaussian distribution. (Several years earlier Wilf 17] found an asymptotic formula for the expected number of distinct part sizes.) This approach culminated in an important paper by Fristedt 9], who proved that the joint distribution of the multiplicities of part sizes is that of independent geometric random variables (Y k), with parameters (1 ? p k), conditioned on the event f P kY k = ng. Fristedt's work, in turn, opened new possibilities and resulted in further progress in our understanding of the structure of random partitions. A good example is a paper of Pittel 16] substantiating two well{known conjectures concerning integer partitions. Utilizing Fristedt's result, quite recently Corteel, Pittel, Savage and Wilf 3] provided an answer to the following question. Consider the following two{step sampling procedure: rst choose uniformly at random a partition of n. Then, out of all di erent part sizes in pick one uniformly at random. What is the asymptotic unconditional probability that this part size has a certain speci ed multiplicity, say, m? For example, partition = f3; 2; 2; 1; 1; 1g of the number 10 has three di erent part sizes 1, 2, and 3 and only one of them has multiplicity three, namely 1. Thus, for this particular partition, the probability of choosing a part that has multiplicity three ...

show abstract

On the multiplicity of parts in a random partition

Cited by 37 publications

References 11 publications

A General Asymptotic Scheme for the Analysis of Partition Statistics

A General Asymptotic Scheme for the Analysis of Partition Statistics

Herbert S. Wilf (1931–2012)

On the Multiplicity of Parts in a Random Composition of a Large Integer

Contact Info

Product

Resources

About