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<p>Let <i>β</i> be an integer and satisfy 0 ≤ <i>β</i> ≤ 5. In this paper, the authors prove that the partition polynomial <Equation ID="Equ1"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11424_2025_4302_Article_Equ1.gif" Format="GIF" Height="49" Rendition="HTML" Resolution="72" Type="Linedraw" Width="167" /> </MediaObject> <EquationSource Format="TEX">\({\prod_{k=1}^n}[1+(2+\beta)q^{k}+q^{2k}]\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <munderover> <mo>∏</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> </mrow> <mo stretchy="false">[</mo> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> <msup> <mi>q</mi> <mrow> <mi>k</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>q</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msup> <mo stretchy="false">]</mo> </math></EquationSource> </Equation> is symmetric and unimodal for <i>n</i> ≥ 1.</p>

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Unimodality of Certain Partition Polynomials

  • 期刊论文

  • 发表日期：

    2025年6月9日

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  • Wanming Guo,
  • Baoxuan Zhu

摘要

Let β be an integer and satisfy 0 ≤ β ≤ 5. In this paper, the authors prove that the partition polynomial \({\prod_{k=1}^n}[1+(2+\beta)q^{k}+q^{2k}]\) ∏ k = 1 n [ 1 + ( 2 + β ) q k + q 2 k ] is symmetric and unimodal for n ≥ 1.