Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">N</mi> </math></EquationSource> </InlineEquation> be the set of all nonnegative integers. For a set <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A\subseteq \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>⊆</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$R_1(A,n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the number of solutions of the equation <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n=a_1+a_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$a_1,a_2\in A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>∈</mo> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation>. Let K be a positive integer, g be an even integer with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$-K\le g/2\le K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi>K</mi> <mo>≤</mo> <mi>g</mi> <mo stretchy="false">/</mo> <mn>2</mn> <mo>≤</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation>, and l be an odd integer with <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$-K\le (l+1)/2\le K+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi>K</mi> <mo>≤</mo> <mo stretchy="false">(</mo> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mn>2</mn> <mo>≤</mo> <mi>K</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we determine the structure of the set <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$A\subseteq \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>⊆</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$R_1(A,2k-1)-R_1(\mathbb {N}\setminus A,2k-1)=g$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mn>2</mn> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <msub> <mi>R</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">N</mi> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mi>A</mi> <mo>,</mo> <mn>2</mn> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>g</mi> </mrow> </math></EquationSource> </InlineEquation> for all integers <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$k\ge K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation> and the structure of the set <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$A\subseteq \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>⊆</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$R_1(A,2k)-R_1(\mathbb {N}\setminus A,2k)=l$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <msub> <mi>R</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">N</mi> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mi>A</mi> <mo>,</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>l</mi> </mrow> </math></EquationSource> </InlineEquation> for all integers <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$k\ge K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation>.

Partitions of natural numbers and their representation function

期刊论文
发表日期：

2026年3月9日

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Xiao-Hui Yan,
Shuang-Shuang Li,
Yu-Mei Li,
Shu-Wen Cheng

摘要

Let $\mathbb {N}$ $N$ be the set of all nonnegative integers. For a set $A\subseteq \mathbb {N}$ $A \subseteq N$ , let $R_1(A,n)$ $R_{1} (A, n)$ be the number of solutions of the equation $n=a_1+a_2$ $n = a_{1} + a_{2}$ with $a_1,a_2\in A$ $a_{1}, a_{2} \in A$ . Let K be a positive integer, g be an even integer with $-K\le g/2\le K$ $- K \leq g / 2 \leq K$ , and l be an odd integer with $-K\le (l+1)/2\le K+1$ $- K \leq (l + 1) / 2 \leq K + 1$ . In this paper, we determine the structure of the set $A\subseteq \mathbb {N}$ $A \subseteq N$ such that $R_1(A,2k-1)-R_1(\mathbb {N}\setminus A,2k-1)=g$ $R_{1} (A, 2 k - 1) - R_{1} (N \ A, 2 k - 1) = g$ for all integers $k\ge K$ $k \geq K$ and the structure of the set $A\subseteq \mathbb {N}$ $A \subseteq N$ such that $R_1(A,2k)-R_1(\mathbb {N}\setminus A,2k)=l$ $R_{1} (A, 2 k) - R_{1} (N \ A, 2 k) = l$ for all integers $k\ge K$ $k \geq K$ .