Partitions and its generalizations are important research objects. For two integers z, j with <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_970_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$1\le j\le z$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>z</mi> </mrow> </math></EquationSource> </InlineEquation>, (z, j)-colored partitions of a positive integer n are those z-colored partitions of n in which at most j colors can appear for a given part size. Let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_970_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$c_{z,j}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>c</mi> <mrow> <mi>z</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the number of (z, j)-colored partitions of n. In this paper, for every prime <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_970_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$p\ge 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>, we determine all positive integers k, l, j, a, b with <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_970_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">$1\le j<p^k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>j</mi> <mo><</mo> <msup> <mi>p</mi> <mi>k</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_970_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$(a, b)=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_970_Article_IEq6.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="188" /> </InlineMediaObject> <EquationSource Format="TEX">$c_{p^k,j}(an+b)\equiv 0\pmod {p^{l}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>c</mi> <mrow> <msup> <mi>p</mi> <mi>k</mi> </msup> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>n</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo>≡</mo> <mn>0</mn> <mspace width="4.44443pt" /> <mrow> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <msup> <mi>p</mi> <mi>l</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all nonnegative integers n.

Congruences for colored partition functions

期刊论文
发表日期：

2025年1月28日

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Wu-Xia Ma,
Yong-Gao Chen

摘要

Partitions and its generalizations are important research objects. For two integers z, j with $1\le j\le z$ $1 \leq j \leq z$ , (z, j)-colored partitions of a positive integer n are those z-colored partitions of n in which at most j colors can appear for a given part size. Let $c_{z,j}(n)$ $c_{z, j} (n)$ be the number of (z, j)-colored partitions of n. In this paper, for every prime $p\ge 5$ $p \geq 5$ , we determine all positive integers k, l, j, a, b with $1\le j<p^k$ $1 \leq j < p^{k}$ and $(a, b)=1$ $(a, b) = 1$ such that $c_{p^k,j}(an+b)\equiv 0\pmod {p^{l}}$ $c_{p^{k}, j} (a n + b) \equiv 0 (mod p^{l})$ for all nonnegative integers n.