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<p>The norm of an integer partition is defined as the product of its parts. This statistic was recently introduced by Schneider in connection to partition zeta functions. In this note, we use the method of moments to study the distribution of the norm under the uniform probability measure on partitions of <i>n</i> as <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_969_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>. We use singularity analysis to prove asymptotics for the moments and show as a result that the norm lacks a non-trivial limiting distribution on <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_969_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\([0,\infty )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

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On the distribution of the norm of partitions

  • 期刊论文

  • 发表日期：

    2025年1月29日

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  • Walter Bridges,
  • William Craig

摘要

The norm of an integer partition is defined as the product of its parts. This statistic was recently introduced by Schneider in connection to partition zeta functions. In this note, we use the method of moments to study the distribution of the norm under the uniform probability measure on partitions of n as \(n \rightarrow \infty \) n → ∞ . We use singularity analysis to prove asymptotics for the moments and show as a result that the norm lacks a non-trivial limiting distribution on \([0,\infty )\) [ 0 , ∞ ) .