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<p>Given an integer partition of <i>n</i> 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 <i>n</i> and prove that, as <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="26_2025_775_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">\(n \rightarrow \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, the sum of reciprocal parts is distributed away from its mean like a random harmonic series.</p>

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Distribution of the Sum of Reciprocal Parts for Distinct Parts Partitions

  • 期刊论文

  • 发表日期：

    2025年7月30日

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  • Walter Bridges

摘要

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 \(n \rightarrow \infty \) n → ∞ , the sum of reciprocal parts is distributed away from its mean like a random harmonic series.