Asymptotics of the number partitioning distribution

We prove an asymptotic formula for the number of partitions of $n$ into distinct parts where the largest part is at most $t\sqrt{n}$ for fixed $t \in \mathbb{R}$. Our method follows a probabilistic approach of Romik, who gave a simpler proof of Szekeres' asymptotic formula for distinct parts partitions when instead the number of parts is bounded by $t\sqrt{n}$. Although equivalent to a circle method/saddle-point method calculation, the probabilistic approach predicts the shap...

The number of solid partitions of a positive integer is an unsolved problem in combinatorial number theory. In this paper, solid partitions are studied numerically by the method of exact enumeration for integers up to 50 and by Monte Carlo simulations using Wang-Landau sampling method for integers up to 8000. It is shown that, for large n, ln[p(n)]/n^(3/4) = 1.79 \pm 0.01, where p(n) is the number of solid partitions of the integer n. This result strongly suggests that the Ma...

We study the asymptotic behaviour of the trace (the sum of the diagonal parts) of a plane partition of the positive integer n, assuming that this parfition is chosen uniformly at random from the set of all such partitions.

The partition function for a system of non-interacting $N-$particles can be found by summing over all the states of the system. The classical partition function for an ideal gas differs from Bosonic or Fermionic partition function in the classical regime. Students find it difficult to follow the differences arising out of incorrect counting by the classical partition function by missing out on the indistinguishability of particles and Fermi-Bose statistics. We present a pedag...

The main aim of this paper is twofold: (1) Suggesting a statistical mechanical approach to the calculation of the generating function of restricted integer partition functions which count the number of partitions --- a way of writing an integer as a sum of other integers under certain restrictions. In this approach, the generating function of restricted integer partition functions is constructed from the canonical partition functions of various quantum gases. (2) Introducing ...

The present article is concerned with the use of approximations in the calculation of the many-body density of states (MBDS) of a system with total energy E, composed by N bosons. In the mean-field framework, an integral expression for MBDS, which is proper to be performed by asymptotic expansions, can be derived. However, the standard second order steepest descent method cannot be applied to this integral when the ground-state is sufficiently populated. Alternatively, we der...

The paper presents a discussion on the asymptotic formula for the number of plane partitions of a large positive integer.

Given an integer partition of $n$ into distinct parts, the sum of the reciprocal parts is an example of an egyptian fraction. We study this statistic under the uniform measure on distinct parts partitions of $n$ and prove that, as $n \to \infty$, the sum of reciprocal parts is distributed away from its mean like a random harmonic sum.

Asymptotic study on the partition function $p(n)$ began with the work of Hardy and Ramanujan. Later Rademacher obtained a convergent series for $p(n)$ and an error bound was given by Lehmer. Despite having this, a full asymptotic expansion for $p(n)$ with an explicit error bound is not known. Recently O'Sullivan studied the asymptotic expansion of $p^{k}(n)$-partitions into $k$th powers, initiated by Wright, and consequently obtained an asymptotic expansion for $p(n)$ along w...

This study extends a prior investigation of limit shapes for partitions of integers, which was based on analysis of sums of geometric random variables. Here we compute limit shapes for grand canonical Gibbs ensembles of partitions of sets, which lead to the sums of Poisson random variables. Under mild monotonicity assumptions, we study all possible scenarios arising from different asymptotic behaviors of the energy, and also compute local limit shape profiles for cases in whi...