Asymptotics of the number partitioning distribution

Recently, there has been renewed interest in studying the asymptotic properties of the number theoretic partition function $p(n)$. Ramanujan, Hardy, and Rademacher provided detailed asymptotic analyses for $p(n)$. Presently, attention has shifted towards Poincar\'e-type asymptotic expansions, characterised by their simplicity albeit reduced accuracy compared to the earlier works of Ramanujan, Hardy, and Rademacher. This paper aims to establish computable error bounds for one ...

The Hardy-Ramanujan partition function asymptotics is a famous result in the asymptotics of combinatorial sequences. It was originally derived using complex analysis and number-theoretic ideas by Hardy and Ramanujan. It was later re-derived by Paul Erd\H{o}s using real analytic methods. Later still, D.J.~Newman used just the usual Hayman saddle-point approach, ubiquitous in asymptotic analysis. Fristedt introduced a probabilistic approach, which was further extended by Dan Ro...

The investigation of partitions of integers plays an important role in combinatorics and number theory. Among the many variations, partitions into powers $0<\alpha<1$ were of recent interest. In the present paper we want to extend our considerations of the length of a random partition by providing a local limit theorem.

For a positive integer $n$, let $p(n)$ be the number of ways to express $n$ as a sum of positive integers. In this note, we revisit the derivation of the Rademacher's convergent series for $p(n)$ in a pedagogical way, with all the details given. We also derive the leading asymptotic behavior of $p(n)$ when $n$ approaches infinity. Some numerical results are tabled.

Fourier expansion of the integrand in the path integral formula for the partition function of quantum systems leads to a deterministic expression which, though still quite complex, is easier to process than the original functional integral. It therefore can give access to problems that eluded solution so far. Here we derive the formula; applications to the problem of Bose-Einstein condensation are presented in the papers arXiv:2108.02659 [math-ph] and arXiv:2208.08931 [math-p...

We study the asymptotic behavior of solid partitions using transition matrix Monte Carlo simulations. If $p_3(n)$ denotes the number of solid partitions of an integer $n$, we show that $\lim_{n\rightarrow\infty} n^{-3/4} \log p_3(n)\sim 1.822\pm 0.001$. This shows clear deviation from the value $1.7898$, attained by MacMahon numbers $m_3(n)$, that was conjectured to hold for solid partitions as well. In addition, we find estimates for other sub-leading terms in $\log p_3(n)$....

The number partitioning problem consists of partitioning a sequence of positive numbers ${a_1,a_2,..., a_N}$ into two disjoint sets, ${\cal A}$ and ${\cal B}$, such that the absolute value of the difference of the sums of $a_j$ over the two sets is minimized. We use statistical mechanics tools to study analytically the Linear Programming relaxation of this NP-complete integer programming. In particular, we calculate the probability distribution of the difference between the c...

Boltzmann sampling is commonly used to uniformly sample objects of a particular size from large combinatorial sets. For this technique to be effective, one needs to prove that (1) the sampling procedure is efficient and (2) objects of the desired size are generated with sufficiently high probability. We use this approach to give a provably efficient sampling algorithm for a class of weighted integer partitions related to Bose-Einstein condensation from statistical physics. Ou...

We study two types of probability measures on the set of integer partitions of $n$ with at most $m$ parts. The first one chooses the random partition with a chance related to its largest part only. We then obtain the limiting distributions of all of the parts together and that of the largest part as $n$ tends to infinity while $m$ is fixed or tends to infinity. In particular, if $m$ goes to infinity not too fast, the largest part satisfies the central limit theorem. The secon...

We study statistical properties of an NP-complete problem, the subset sum, using the methods and concepts of statistical mechanics. The problem is a generalization of the number partitioning problem, which is also an NP-complete problem and has been studied in the physics literature. The asymptotic expressions for the number of solutions are obtained. These results applied to the number partitioning problem as a special case are compared with those which were previously obtai...