Chang and Chen (J Stat Phys 131(4):631–650, 2008) and Chang et al. (J Stat Phys 126(3):649–667, 2007) present the number of dimer coverings and spanning trees on the Sierpinski gasket <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2024_3392_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$SG_b(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>G</mi> <mi>b</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> at stage n with the side length b equal to two, three and four, respectively. In this paper, we obtain the exact closed formula of the number of dimer coverings and spanning trees on the silicate-type Sierpinski gasket <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2024_3392_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$SO_b(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>O</mi> <mi>b</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> at stage n with the side length <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2024_3392_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">$b=2,3,4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>.

On the Statistics of Dimer Coverings and Spanning Trees on the Silicate-Type Sierpinski Gasket

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发表日期：

2025年1月20日

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Xingsheng Yang,
Jingchao Lai,
Weigen Yan

摘要

Chang and Chen (J Stat Phys 131(4):631–650, 2008) and Chang et al. (J Stat Phys 126(3):649–667, 2007) present the number of dimer coverings and spanning trees on the Sierpinski gasket $SG_b(n)$ $S G_{b} (n)$ at stage n with the side length b equal to two, three and four, respectively. In this paper, we obtain the exact closed formula of the number of dimer coverings and spanning trees on the silicate-type Sierpinski gasket $SO_b(n)$ $S O_{b} (n)$ at stage n with the side length $b=2,3,4$ $b = 2, 3, 4$ .