Meinardus' theorem on weighted partitions: extensions and a probabilistic proof

We give a probalistic proof of the famous Meinardus' asymptotic formula for the number of weighted partitions with weakened one of the three Meinardus' conditions, and extend the resulting version of the theorem to other two classis types of decomposable combinatorial structures, which are called assemblies and selections. The results obtained are based on combining Meinardus' analytical approach with probabilistic method of Khitchine.Comment: The version contains a few minor corrections.It will be published in Advances in Applied Mathematic

A Phase transition in the distribution of the length of integer partitions

Paper presented at Strathmore International Math Research Conference on July 23 - 27, 2012We assign a uniform probability to the set consisting of partitions of a positive integer n such that the multiplicity of each summand is less than a given number d and we study the limiting distribution of the number of summands in a random partition. It is known from a result by Erd˝os and Lehner published in 1941 that the distributions of the length in random restricted (d = 2) and random unrestricted (d n + 1) partitions behave very differently. In this paper we show that as the bound d increases we observe a phase transition in which the distribution goes from the Gaussian distribution of the restricted case to the Gumbel distribution of the unrestricted case.We assign a uniform probability to the set consisting of partitions of a positive integer n such that the multiplicity of each summand is less than a given number d and we study the limiting distribution of the number of summands in a random partition. It is known from a result by Erd˝os and Lehner published in 1941 that the distributions of the length in random restricted (d = 2) and random unrestricted (d n + 1) partitions behave very differently. In this paper we show that as the bound d increases we observe a phase transition in which the distribution goes from the Gaussian distribution of the restricted case to the Gumbel distribution of the unrestricted case

The asymptotic number of weighted partitions with a given number of parts

For a given sequence $b_k$ of non-negative real numbers, the number of weighted partitions of a positive integer $n$ having $m$ parts $c_{n,m}$ has bivariate generating function equal to $\prod_{k=1}^\infty (1-yz^k)^{-b_k}$. Under the assumption that $b_k\sim Ck^{r-1}$, $r>0$, and related conditions on the Dirichlet generating function of the weights $b_k$, we find asymptotics for $c_{n,m}$ when $m=m(n)$ satisfies $m=o\left(n^\frac{r}{r+1}\right)$ and \lim_{n\to\infty}m/\log^{3+\eps}n=\infty, \eps>0

A phase transition in the distribution of the length of integer partitions

We assign a uniform probability to the set consisting of partitions of a positive integer $n$ such that the multiplicity of each summand is less than a given number $d$ and we study the limiting distribution of the number of summands in a random partition. It is known from a result by Erdős and Lehner published in 1941 that the distributions of the length in random restricted $(d=2)$ and random unrestricted $(d \geq n+1)$ partitions behave very differently. In this paper we show that as the bound $d$ increases we observe a phase transition in which the distribution goes from the Gaussian distribution of the restricted case to the Gumbel distribution of the unrestricted case