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<p>We prove specific biases in the number of occurrences of parts belonging to two different residue classes <i>a</i> and <i>b</i>, modulo a fixed nonnegative integer <i>m</i>, for the sets of unrestricted partitions, partitions into distinct parts, and overpartitions. These biases follow from inequalities for residue-weighted partition functions for the respective sets of partitions. We also establish asymptotic formulas for the numbers of partitions of size <i>n</i> that belong to these sets of partitions and have a symmetric residue class bias (i.e., for <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40687_2025_502_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="95" /> </InlineMediaObject> <EquationSource Format="TEX">\(1\le a&lt;m/2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>a</mi> <mo>&lt;</mo> <mi>m</mi> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40687_2025_502_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">\(b=m-a\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>=</mo> <mi>m</mi> <mo>-</mo> <mi>a</mi> </mrow> </math></EquationSource> </InlineEquation>), as <i>n</i> tends to infinity.</p>

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Residue class biases in unrestricted partitions, partitions into distinct parts, and overpartitions

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  • 发表日期：

    2025年2月21日

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  • Michael J. Schlosser,
  • Nian Hong Zhou

摘要

We prove specific biases in the number of occurrences of parts belonging to two different residue classes a and b, modulo a fixed nonnegative integer m, for the sets of unrestricted partitions, partitions into distinct parts, and overpartitions. These biases follow from inequalities for residue-weighted partition functions for the respective sets of partitions. We also establish asymptotic formulas for the numbers of partitions of size n that belong to these sets of partitions and have a symmetric residue class bias (i.e., for \(1\le a<m/2\) 1 ≤ a < m / 2 and \(b=m-a\) b = m - a ), as n tends to infinity.