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<p>We present another combinatorial proof of a result by Garvan and Jennings-Shaffer regarding the characterization of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\overline{\textrm{spt}}\hspace{0.55542pt}(n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mtext>spt</mtext> <mo>¯</mo> </mover> <mspace width="0.55542pt" /> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\overline{\operatorname {spt}}_1(n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover> <mo>spt</mo> <mo>¯</mo> </mover> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\overline{\operatorname {spt}}_2(n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover> <mo>spt</mo> <mo>¯</mo> </mover> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> modulo 2 without using the concept of the crank of a partition.</p>

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Another combinatorial proof of a result by Garvan and Jennings-Shaffer related to \(\overline{\textrm{spt}}\hspace{0.55542pt}(n)\)

  • 期刊论文

  • 发表日期：

    2026年2月11日

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  • Suparno Ghoshal,
  • Arijit Jana,
  • Imdadul Hussain

摘要

We present another combinatorial proof of a result by Garvan and Jennings-Shaffer regarding the characterization of \(\overline{\textrm{spt}}\hspace{0.55542pt}(n)\) spt ¯ ( n ) , \(\overline{\operatorname {spt}}_1(n)\) spt ¯ 1 ( n ) , and \(\overline{\operatorname {spt}}_2(n)\) spt ¯ 2 ( n ) modulo 2 without using the concept of the crank of a partition.