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Distribution of the sum of reciprocal parts for distinct parts partitions

Walter Bridges

TL;DR

This work studies the sum of reciprocals of distinct parts in partitions of $n$ under the uniform measure. The authors decompose the reciprocal-sum into small, intermediate, and large part ranges, prove that the small-range contribution converges in distribution to a random harmonic sum $H=\sum_{k\ge1}\frac{\varepsilon_k}{k}$, and show that the intermediate and large ranges contribute vanishingly to the fluctuations around the mean. Leveraging Fristedt-type limit shapes and elementary arguments, they establish the limit $\lim_{n\to\infty} P_n\left(2S-\log(\sqrt{3n})\le x\right)=P(H\le x)$, connecting Egyptian fractions with a probabilistic harmonic sum. The results illuminate the distributional behavior of distinct-part Egyptian fractions and provide an accessible, shape-informed route to a nontrivial limit law for $S$.

Abstract

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.

Distribution of the sum of reciprocal parts for distinct parts partitions

TL;DR

This work studies the sum of reciprocals of distinct parts in partitions of under the uniform measure. The authors decompose the reciprocal-sum into small, intermediate, and large part ranges, prove that the small-range contribution converges in distribution to a random harmonic sum , and show that the intermediate and large ranges contribute vanishingly to the fluctuations around the mean. Leveraging Fristedt-type limit shapes and elementary arguments, they establish the limit , connecting Egyptian fractions with a probabilistic harmonic sum. The results illuminate the distributional behavior of distinct-part Egyptian fractions and provide an accessible, shape-informed route to a nontrivial limit law for .

Abstract

Given an integer partition of 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 and prove that, as , the sum of reciprocal parts is distributed away from its mean like a random harmonic sum.

Paper Structure

This paper contains 6 sections, 5 theorems, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Contribution from small parts
  3. Contribution from intermediate parts
  4. Contribution from large parts
  5. Proof of Theorem \ref{['T:Main']}
  6. Strong limit shape

Key Result

Theorem 1.1

For any $x \in \mathbb{R}$, we have

Figures (2)

  • Figure 1: A histogram of 10 000 values of $2S(\lambda)-\log(\sqrt{3|\lambda|})$, where partitions $\lambda$ have been generated in Maple by a Boltzmann sampler with parameter $q=e^{-\frac{\pi}{\sqrt{12n}}}$ with $n=2000$ (see DFLSFristedt). In red is an approximation to the density for $H$, as described in Schmuland.
  • Figure 2: The black step functions are the renormalized shapes $\frac{1}{\sqrt{|\lambda|}}\phi_{\lambda}(\sqrt{|\lambda|}t)$ for six random distinct parts partitions $\lambda$ of sizes 992, 1592, 1065, 1475, 910, and 1107, generated using a Boltzmann sampler with parameter $q=e^{-\frac{\pi}{\sqrt{12n}}}$ with $n=1000$ (see DFLSFristedt). In red are the curves $L(t) \pm n^{-\frac{1}{4}}$.

Theorems & Definitions (9)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • proof
  • Proposition A.1
  • proof