A central limit theorem for integer partitions

Madritsch, Manfred G.; Wagner, Stephan

doi:10.1007/s00605-009-0126-y

Cited by 6 publications

(8 citation statements)

References 18 publications

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“…Remark The asymptotic formula we get in (19) is very similar to those we find in [12] or [14] and therefore, the following bounds hold for the tails…”

Section: Propositionsupporting

confidence: 56%

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On the number of summands in a random prime partition

Ralaivaosaona

2011

Monatsh Math

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We study the length (number of summands) in partitions of an integer into primes, both in the restricted (unequal summands) and unrestricted case. It is shown how one can obtain asymptotic expansions for the mean and variance (and potentially higher moments), which is in contrast to the fact that there is no asymptotic formula for the number of such partitions in terms of elementary functions. Building on ideas of Hwang, we also prove a central limit theorem in the restricted case. The technique also generalizes to partitions into powers of primes, or even more generally, the values of a polynomial at the prime numbers.

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“…Remark The asymptotic formula we get in (19) is very similar to those we find in [12] or [14] and therefore, the following bounds hold for the tails…”

Section: Propositionsupporting

confidence: 56%

“…Efforts have been made to reduce these conditions, see for instance [14], but yet the sequence of primes fails to satisfy these conditions, as also mentioned explicitly in Hwang's paper.…”

mentioning

confidence: 90%

On the number of summands in a random prime partition

Ralaivaosaona

2011

Monatsh Math

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“…. Finally, we note that an interesting extension of Meinardus' theorem in a related but a quite different direction was obtained in [8] and [9]. In both papers it is studied the problem of asymptotic enumeration of the number of unweighted partitions (b k = 1, k ≥ 1) of n with summands belonging to a given infinite set of positive integers Λ = {λ 1 , .…”

Section: Statement History and Motivation Of The Main Resultsmentioning

confidence: 91%

A Meinardus Theorem with Multiple Singularities

Granovsky

Stark

2012

Commun. Math. Phys.

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Meinardus proved a general theorem about the asymptotics of the number of weighted partitions, when the Dirichlet generating function for weights has a single pole on the positive real axis. Continuing [3], we derive asymptotics for the numbers of three basic types of decomposable combinatorial structures (or, equivalently, ideal gas models in statistical mechanics) of size n, when their Dirichlet generating functions have multiple simple poles on the positive real axis. Examples to which our theorem applies include ones related to vector partitions and quantum field theory. Our asymptotic formula for the number of weighted partitions disproves the belief accepted in the physics literature that the main term in the asymptotics is determined by the rightmost pole.

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“…These conditions are used by Hwang [31], for example, where the distribution of the number of summands (taken without multiplicities) follows a central and local limit theorem. There is also a central limit theorem proved by Madritsch and Wagner in [38], concerning partitions whose base-b representation only contains digits from some given set.…”

Section: More General Growth Conditionsmentioning

confidence: 96%