In this paper, we examine the functions <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1109_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$P_k(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>P</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which counts the partitions of n into exactly k parts, and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1109_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$Q_k(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>Q</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which counts the partitions of n into exactly k distinct parts. These partition functions are closely linked to two classical identities of Euler. We explore this connection and establish several new relationships between <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1109_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$P_k(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>P</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1109_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$Q_k(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>Q</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We give both analytic and combinatorial proofs of the theorems.

Partitions into an exact number of parts

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2025年5月23日

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Cristina Ballantine,
Mircea Merca

摘要

In this paper, we examine the functions $P_k(n)$ $P_{k} (n)$ , which counts the partitions of n into exactly k parts, and $Q_k(n)$ $Q_{k} (n)$ , which counts the partitions of n into exactly k distinct parts. These partition functions are closely linked to two classical identities of Euler. We explore this connection and establish several new relationships between $P_k(n)$ $P_{k} (n)$ and $Q_k(n)$ $Q_{k} (n)$ . We give both analytic and combinatorial proofs of the theorems.