[Paper Review] A George Szekeres Formula for Restricted Partitions

L. Bruce Richmond|arXiv (Cornell University)|Mar 22, 2018

Advanced Combinatorial Mathematics11 references3 citations

TL;DR

This paper derives an asymptotic formula for the number of integer partitions of $ n $ into at most $ j $ parts, each at most $ r $, using the George Szekeres circle method and saddlepoint techniques. It shows that the probability a random partition of an even integer $ n $ is graphical—i.e., corresponds to the degree sequence of a simple graph—is $ O(\ln^{-1/2}n) $, with independence of rank distributions and precise error bounds via probabilistic and analytic number theory tools.

ABSTRACT

We derive an asymptotic formula for $A(n,j,r)$ the number of integer partitions of $n$ into at most $j$ parts each part $\le r$. We assume $j$ and $r$ are near their mean values. We also investigate the second largest part, the number of parts $\ge 2$, etc. We show that the fraction of the partitions of an even integer $n$ that are graphical, ie. whose parts form the degree sequence of a simple graph, is $O(\ln^{-1/2} n)$. Probabilistic results are used in our discussion of graphical partitions. The George Szekeres circle method is essential for our asymptotic results on partitions. We determine the distributions defined by the successive ranks of partitions, generalizing the result of Erdos and Richmond for the rank of a partition.

Motivation & Objective

• To derive an asymptotic formula for $ A(n,j,r) $, the number of integer partitions of $ n $ into at most $ j $ parts, each $ \leq r $, when $ j $ and $ r $ are near their mean values.
• To analyze the distribution of successive ranks in integer partitions, generalizing Erdős and Richmond’s result on the rank of a partition.
• To investigate the probability that a random partition of an even integer $ n $ is graphical, i.e., its parts form a degree sequence of a simple graph.
• To refine error estimates in the saddlepoint method for partition generating functions, achieving $ O(n^{-1/7}) $ error terms.
• To establish the asymptotic independence of the largest part and the number of parts in typical partitions.

Proposed method

• Uses the generating function $ \sum_{n=0}^\infty A(n,j,r)x^n = \prod_{\nu=1}^r \frac{1 - x^{j+\nu}}{1 - x^\nu} $, which is the Gauss binomial coefficient.
• Applies the Szekeres circle method, expressing $ A(n,j,r) $ as a contour integral over a circle of radius $ e^{-\alpha} $ with $ \alpha = \pi / \sqrt{6n} $.
• Employs the saddlepoint method to approximate the integral, expanding the logarithm of the integrand in powers of $ \theta $, and estimates the main contribution over $ |\theta| \leq \theta_0 = n^{-5/7} $.
• Uses Cauchy’s theorem and asymptotic analysis of the second and third derivatives of the phase function to control error terms, showing the tail integral is negligible.
• Applies Esseen’s inequality to bound the tail probability of a sum of independent rank-distributed random variables, leading to the graphical partition bound.
• Derives the density of the $ k $-th successive rank $ r_k(t) $ using gamma functions and Laplace transform techniques, showing independence of rank distributions.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of $ A(n,j,r) $, the number of partitions of $ n $ into at most $ j $ parts each $ \leq r $, when $ j $ and $ r $ are near their mean values?
• RQ2 How are the distributions of the successive ranks of integer partitions related, and can they be shown to be independent?
• RQ3 What is the probability that a random partition of an even integer $ n $ is graphical, i.e., its parts form a degree sequence of a simple graph?
• RQ4 Can the error term in the saddlepoint approximation for $ A(n,j,r) $ be bounded more precisely, specifically as $ O(n^{-1/7}) $?
• RQ5 Does the distribution of the largest part in a partition become independent of the number of parts in the asymptotic regime?

Key findings

• The number of partitions of $ n $ with $ r_k(t) = \lfloor \sqrt{6n}/\pi \cdot t_k \rfloor $ is asymptotically $ P(n) \cdot \frac{\pi}{\sqrt{6n}} \cdot \frac{\Gamma(2k)}{\Gamma^2(k)} \cdot \left(1 + e^{-t_k}\right)^{-2k} \left(1 + e^{t_k}\right)^{-2k} $.
• The distributions of the successive ranks $ r_k(t) $ are mutually independent, allowing joint asymptotic analysis for finitely many $ k $.
• The variance of the $ k $-th rank distribution satisfies $ \sigma_k^2 \sim \pi k^{-1}/2 $, and the third absolute moment $ \rho_k \sim 12k^{-3/2}\sqrt{\pi} $.
• The sum of the rank distributions has a tail probability bounded by $ O(\ln^{-1/2}n) $, leading to the conclusion that the probability a random partition of an even $ n $ is graphical is $ O(\ln^{-1/2}n) $.
• The error term in the saddlepoint approximation for $ A(n,j,r) $ is $ O(n^{-1/7}) $, improving upon earlier estimates.
• The largest part and the number of parts in a typical partition are asymptotically independent, confirming a result not previously available in the literature.