The minimal excludant of a partition <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="26_2025_773_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> is the smallest positive integer that is not a part of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="26_2025_773_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>. For <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="26_2025_773_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, Shen proved that the number of partitions of n with the minimal excludant being odd is always greater than or equal to the number of partitions of n with the minimal excludant being even. Later, Andrews–Newman and Hopkins–Sellers independently discovered that the number of partitions of n in which the minimal excludant is odd (resp. even) equals the number of partitions of n with non-negative (resp. positive) crank, where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="26_2025_773_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we find a new refinement of the aforementioned result, and establish two interesting partition inequalities, one of which generalizes Shen’s result.

Identities and Inequalities Involving the Crank and Minimal Excludant of Integer Partitions

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2025年7月13日

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Bernard L. S. Lin,
Han Liu

摘要

The minimal excludant of a partition $\lambda $ $λ$ is the smallest positive integer that is not a part of $\lambda $ $λ$ . For $n\ge 2$ $n \geq 2$ , Shen proved that the number of partitions of n with the minimal excludant being odd is always greater than or equal to the number of partitions of n with the minimal excludant being even. Later, Andrews–Newman and Hopkins–Sellers independently discovered that the number of partitions of n in which the minimal excludant is odd (resp. even) equals the number of partitions of n with non-negative (resp. positive) crank, where $n\ge 2$ $n \geq 2$ . In this paper, we find a new refinement of the aforementioned result, and establish two interesting partition inequalities, one of which generalizes Shen’s result.