The A-partition function <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1048_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$p_A(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mi>A</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> enumerates those partitions of n whose parts belong to a fixed (finite or infinite) set A of positive integers. On the other hand, the extended A-partition function <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1048_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$p_A\left( \varvec{\mu }\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mi>A</mi> </msub> <mfenced close=")" open="("> <mrow> <mi mathvariant="bold-italic">μ</mi> </mrow> </mfenced> </mrow> </math></EquationSource> </InlineEquation> is defined as an multiplicative extension of the A-partition function to a function on A-partitions. In this paper, we investigate the Bessenrodt–Ono type inequality for a wide class of A-partition functions. In particular, we examine the property for both the m-ary partition function <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1048_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$b_m(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>b</mi> <mi>m</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and the d-th power partition function <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1048_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$p_d(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mi>d</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Moreover, we show that <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1048_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$b_m(\varvec{\mu })$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>b</mi> <mi>m</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="bold-italic">μ</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1048_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$p_d(\varvec{\mu })$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mi>d</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="bold-italic">μ</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>) takes its maximum value at an explicitly described set of m-ary partitions (power partitions), where <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1048_Article_IEq7.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\varvec{\mu }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">μ</mi> </mrow> </math></EquationSource> </InlineEquation> is an m-ary partition (a power partition) of n. Additionally, we exhibit analogous results for the Fibonacci partition function and the ‘factorial’ partition function. It is worth pointing out that an elementary combinatorial reasoning plays a crucial role in our investigation.

On the Bessenrodt–Ono type inequality for a wide class of A-partition functions

期刊论文
发表日期：

2025年2月25日

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Krystian Gajdzica

摘要

The A-partition function $p_A(n)$ $p_{A} (n)$ enumerates those partitions of n whose parts belong to a fixed (finite or infinite) set A of positive integers. On the other hand, the extended A-partition function $p_A\left( \varvec{\mu }\right) $ $p_{A} (μ)$ is defined as an multiplicative extension of the A-partition function to a function on A-partitions. In this paper, we investigate the Bessenrodt–Ono type inequality for a wide class of A-partition functions. In particular, we examine the property for both the m-ary partition function $b_m(n)$ $b_{m} (n)$ and the d-th power partition function $p_d(n)$ $p_{d} (n)$ . Moreover, we show that $b_m(\varvec{\mu })$ $b_{m} (μ)$ ( $p_d(\varvec{\mu })$ $p_{d} (μ)$ ) takes its maximum value at an explicitly described set of m-ary partitions (power partitions), where $\varvec{\mu }$ $μ$ is an m-ary partition (a power partition) of n. Additionally, we exhibit analogous results for the Fibonacci partition function and the ‘factorial’ partition function. It is worth pointing out that an elementary combinatorial reasoning plays a crucial role in our investigation.